The Catholic Church professes a belief in the existence of limbo. Although the doctrine on this subject has not yet been defined, does the Catholic Church believe that it is eternal?
We must distinguish between the two limbos:
- Limbo of hell or of the Patriarchs (limbus inferni seu patrum)
This is also known as Abraham's bosom. This limbo no longer exists (cf. "Reply to Objection 3" of this) because Christ has already descended into hell and brought those souls detained there to heaven during His Ascension.
- Limbo for children (limbus puerorum)
This is different than the limbo of the Fathers, which "is placed higher than the limbo of children." (cf. "I answer that…" of this). It is where the souls of unbaptized children are. It is a place of perpetual, natural happiness where souls are not actively punished by fire because they have not committed any personal sins; however, it is still hell, which, by definition, is the place where one is deprived of the Beatific Vision.
As St. Thomas Aquinas writes ("I answer that…" of this):
The limbo of the Fathers and the limbo of children, without any doubt, differ as to the quality of punishment or reward. For children have no hope of the blessed life, as the Fathers in limbo had, in whom, moreover, shone forth the light of faith and grace. But as regards their situation, there is reason to believe that the place of both is the same; except that the limbo of the Fathers is placed higher than the limbo of children, just as we have stated in reference to limbo and hell (Article ).
Now, hell is eternal. The limbus puerorum is in hell. Therefore, the limbus puerorum is eternal.
Pope Pius VI condemned "as a Pelagian fable" the fact that the false Synod of Pistoia "introduced that middle place and state free of guilt and of punishment between the kingdom of God and eternal damnation.
The full quote of Pope Pius VI's “Auctorem fidei,” Aug. 28, 1794, is reproduced in Denzinger-Schönmetzer, Enchiridion Symbolorum, definitionum et declarationum de rebus fidei et morum. Ed. 34. 1965, §2626
- The doctrine which rejects as a Pelagian fable, that place of the lower regions (which the faithful generally designate by the name of the limbo of children) in which the souls of those departing with the sole guilt of original sin are punished with the punishment of the condemned, exclusive of the punishment of fire, just as it, by this very fact, that these who remove the punishment of fire introduced that middle place and state free of guilt and of punishment between the kingdom of God and eternal damnation, such as that about which the Pelagians idly talk,—false, rash, injurious to Catholic schools.
Where do these souls end up, then? They can't be in heaven or purgatory because they are not baptized, and they can't be punished with the fires of hell because they committed no personal sin; nor can they be somewhere between heaven and hell. Thus, they must be in hell (because they are deprived of the Beatific Vision, and hell is the place of being deprived of seeing God), but they are not punished by the fires of hell. This is the limbus puerorum.
1. Distinguishing between two limbos
In Catholic doctrine, there are two states or "places" of the souls of dead people which are called "limbo":
The Limbo of Infants (Latin limbus infantium or limbus puerorum), inhabited by the souls of those who die in original sin only, where they do not suffer any torment, do not see God, and do not have any expectation of being redeemed and taken to the vision of God at some future time. It is de fide doctrine that this is an everlasting state for each soul in it , and consequently that, as a potential state for souls or "place", it existed before the death of Christ, exists after it, and will exist forever. (But it is not de fide doctrine that there are actually any souls in it, because it is not de fide doctrine that any person actually dies in original sin only, see section 3.)
The "Bosom of Abraham", "Limbo of the Patriarchs" or "Limbo of the Fathers" (Latin limbus patrum), inhabited by the souls of the purified righteous before Christ's death, where they did not suffer any torment, did not see God, but were comforted by the expectation that at some future time they would be redeemed and taken to the vision of God (Lk 16:22). It is de fide doctrine that this was a temporary state for each soul in it, which ended after Christ's death, and that, as a potential state for souls or "place", it existed only before the death of Christ .
2. Magisterial definitions about the fate of those who die in original sin only
The doctrine that those who die in original sin only go to an everlasting state of privation of the Beatific Vision, commonly called limbo of infants, is de fide, having been taught in the following magisterial definitions . Note that the most authoritative of those definitions, i.e. those of Ecumenical Councils, do not state explicitely that souls in the limbo of infants do not suffer any torment of fire, so that this specific point is not de fide but only sententia communis.
Pope Innocent III, letter "Maiores Ecclesiae causas" to Imbert, archbishop of Arles, 1201, DS 780 Dz 410:
The punishment of original sin is deprivation of the vision of God, but the punishment of actual sin is the torments of everlasting hell.
Ecumenical Council of Lyon II, Profession of Faith of Michael Palaeologus, 1274, DS 858 Dz 464:
The souls of those who die in mortal sin or with original sin only, however, immediately descend to hell, yet to be punished with different punishments.
Ecumenical Council of Florence, Session 6 — 6 July 1439, promulgated as bull "Laetentur Caeli: Bulla Unionis Graecorum" by Pope Eugene IV, DS 1306 Dz 693:
Moreover, the souls of those who depart in actual mortal sin or in original sin only, descend immediately into hell, yet to be punished with different punishments.
Illorum autem animas, qui in actuali mortali peccato vel solo originali decedunt, mox in infernum descendere, penis tamen disparibus puniendas. 
Pope Pius VI, Constitution "Auctorem fidei," 28 Aug. 1794, listing condemned propositions of the Synod of Pistoia, DS 2626 Dz 1526:
- The doctrine which rejects as a Pelagian fable, that place of the lower regions (which the faithful generally designate by the name of the limbo of children) in which the souls of those departing with the sole guilt of original sin are punished with the punishment of the condemned, exclusive of the punishment of fire, just as if, by this very fact, (that) these who remove the punishment of fire introduced that middle place and state free of guilt and of punishment between the kingdom of God and eternal damnation, such as that about which the Pelagians idly talk,--false, rash, injurious to Catholic schools.
Since the original of the text in italics is perinde ac si hoc ipso quod qui poenam ignis removent, the "that" in the usual English translation does not belong.
Note that the Pistoians did not hold that limbo, not featuring fire, was a "middle place and state free of guilt and of punishment between the kingdom of God and eternal damnation", but on the contrary, accused those who held that limbo did not feature fire of holding that. That is, the Pistoians held that the limbo of infants did feature fire and rejected as a Pelagian fable the (wholly legitimate) position that it did not.
3. Whether infants who die unbaptized die in original sin
It is evident that, in order to state positively that an infant who dies upbaptized dies in a state of original sin, it is necessary to state positively that God, in the last second of the infant's life, does not reveal Himself and his love directly to the soul of the dying infant and asks him or her for a response to that revelation, so that if the infant accepts God's love and turns to God, it is a case of baptism of desire whereby God infuses sanctifying grace and charity to the soul of the infant and he or she goes to the Beatific Vision.
The hypothesis that God performs that revelation in the last instant of a dying infant's life is called "illumination theory", and while it is obvious that it cannot be affirmed that this is the actual case, it is also obvious that it cannot be affirmed that this is not the actual case .
The possible objection that the last second of an infant's life is too short a time for the divine revelation and the infant's response can be addressed very easily. It is well known that, according to ordinary laws of physics, time flows at different "speeds" in different parts of the universe, e.g. those where the gravitational field has different strength (which is taken into account in the GPS system). From that, it is easy to see that God can change the time scale of the child's soul with respect to the rest of the universe, so that a whole day elapses for the soul of the dying child while only a second elapses in the rest of the universe.
Moreover, that was probably the case when the sun stood still for a day in Gibeon at the order of Joshua (Josh 10:12-14): God did not stop the Earth's rotation (or the sun's movement, for the geocentrics out there), but changed the scale of time flow in the battlefield 100,000 to 1 with respect to the rest of the universe. Why didn't God just kill all the Amorites in a flash? The message of the passage is loud and clear: if God wants the collaboration of his chosen ones in order that they win a battle, He will give them the time they need for that collaboration, even if it requires changing the scale of their time flow 100,000 to 1 with respect to the rest of the universe.
Finally, it should be mentioned that one of the foremost orthodox Catholic theologians of the beginning of the XX century, Cardinal Louis Billot SJ, in a series of articles published in "Etudes religieuses" in 1919-1925, proposed the hypothesis that a large number of chronologically adult persons remain, in the theological-moral plane, in the same situation as that of children before attaining the use of reason, because they have attained the use of reason with respect to the practical ends of life but not with respect to the ultimate end, so that they have not had the possibility to know God and thus turn to Him or reject Him, and that therefore the state of those adults after death is also the limbo of infants (unless, of course, the illumination hypothesis is also right).
 And, after the resurrection of the dead, for each whole human person, soul and body, in it.
 Pope Benedict XII in his 1336 Apostolic Constitution "Benedictus Deus" [2.a] [2.b] defined that the souls of all the righteous who were completely purified at the time of Christ's death were taken to the Beatific Vision at that time, and that after that time, the souls of all the rigtheous, once they are completely purified, go directly to the Beatific Vision. From that, it follows that, after the death of Christ, the "Bosom of Abraham" no longer exists as a potential state for souls or "place".
 DS = Denzinger-Schönmetzer; Dz = Denzinger.
[3.a] Denzinger-Schönmetzer 1963: http://www.clerus.org/bibliaclerusonline/en/lt.htm
[3.b] Denzinger 1954: http://patristica.net/denzinger/
 Brian W. Harrison, O.S., 2005, "Could Limbo Be 'Abolished'?". http://www.seattlecatholic.com/a051207.html
The article, which does a good review of Catholic doctrine, says about the illumination theory:
"Are we to suppose that God miraculously 'fast-forwards' the mental development of these infants (and gravely retarded persons) in the instant before death, following this up with a special illumination so as to render them capable of an at least implicit desire for baptism? But miracles cannot be gratuitously postulated, so we could never be sure, in the absence of any revealed truth in Scripture or Tradition, that this is in fact what God does."
The obvious reply is: "Are we instead to suppose that God does not do that?" Because just as we cannot be sure that this is in fact what God does, we cannot either be sure that this is not in fact what God does.
The concept of limbo probably began with Thomas Aquinas but for some reason was never formalised as Catholic doctrine. When I was a young child, many years ago, the brothers taught me about limbo as fact, and I was certainly required to be able answer test questions on the subject, but it seems it never had formal doctrinal status.
Recent Catholic theological speculation tends to stress the hope, though not the certainty, that unbaptised infants may attain heaven instead of the supposed state of limbo. Since Vatican II, the Catholic Church has begun to move away from its former teachings about limbo and treat it as no more than a hypothesis, and there have been suggestions that after the passage of sufficient time, the hypothesis will be abandoned. Now, the International Theological Commission says:
It is clear that the traditional teaching on this topic has concentrated on the theory of limbo, understood as a state which includes the souls of infants who die subject to original sin and without baptism, and who, therefore, neither merit the beatific vision, nor yet are subjected to any punishment, because they are not guilty of any personal sin. This theory, elaborated by theologians beginning in the Middle Ages, never entered into the dogmatic definitions of the Magisterium, even if that same Magisterium did at times mention the theory in its ordinary teaching up until the Second Vatican Council. It remains therefore a possible theological hypothesis. However, in the Catechism of the Catholic Church (1992), the theory of limbo is not mentioned.
Limbo was never intended to be eternal, simply a half-way station until the Last Judgement, at which time these souls would be admitted to heaven. I am sure the Catholic Church would now say it does not have a doctrinal position on this.
Limbo falls into a Category which in the Catholic Church is call "THEOLOGOUMENON"
In Roman Catholic Theology, is usually refers to statements that are without direct confirmation in Scripture or official endorsement by the divinely inspired Teaching Magisterium of the Church, and are therefore not dogmatically binding per se, but are worth recommending because they cast light on understanding doctrines that are considered to be divinely revealed. wiki
Limbo has always fallen into the category of THEOLOGOUMNON. Presently, they idea of limbo is not accepted or taught as the concept of Gods perfect Mercy towards children, is more favorable considering his perfect nature.
Other THEOLOGOUMON in Catholicism are the fiery torments and countless years in purgatory. As Theologians look closer at this mystery and compare it to the perfect mercy of God, they start to see that Purgatory is not so much a place but a state, not a time but an experience. The reality of our final sanctification is still Theologoumenon.
I can say with certainty that as a Catholic in the current age, LIMBO has never been taught to me as it no longer applies in our current environment where the unborn are murdered in the womb.
Furthermore, the idea of limbo brings negative emotional reactions to especially non-catholics, when the teachings of the Church should be, Mercy to the victims, not pain and suffering. Limbo is often used by anti-catholics as they persecute the Church, usually by misrepresenting it's teaching, and in so doing, persecute Christ by being false witnesses to it.