Catholics reconcile the two beliefs by being allowed to believe in evolution, but required to believe in the existence of Adam and Eve.
For the purposes of this discussion, evolution is the scientific hypothesis that the physical bodies of various living beings have developed from those of other living beings of different species. To believe in evolution is to hold evolution as a scientific hypothesis, subject to falsifiability in the event of countering evidence (that is, evidence that can only be explained if the hypothesis is false).
With these definitions in mind, then, the Catholic Church neither requires nor forbids Catholics to believe in evolution—to the extent (and only to the extent) that such a belief is not found to conflict with Church teaching. However, the Church does require that Catholics assert that there were a single original man and woman, who sinned, and who are the ancestors of all subsequent humans.
The encyclical letter Humani Generis, written by Pope Pius XII in 1950, discusses (among other philosophical stances the Church considers to be errors or heresies) evolution as it has sometimes been used to back an existential, relativist view of the world. Pope Pius writes:
The Teaching Authority of the Church does not forbid that, in conformity with the present state of human sciences and sacred theology, research and discussions, on the part of men experienced in both fields, take place with regard to the doctrine of evolution, in as far as it inquires into the origin of the human body as coming from pre-existent and living matter—for the Catholic faith obliges us to hold that souls are immediately created by God. However, this must be done in such a way that the reasons for both opinions, that is, those favorable and those unfavorable to evolution, be weighed and judged with the necessary seriousness, moderation and measure, and provided that all are prepared to submit to the judgment of the Church, to whom Christ has given the mission of interpreting authentically the Sacred Scriptures and of defending the dogmas of faith.
In other words, the Church has no issue with discussion of evolution as science, and purely as science; it's when the scientific conclusions are used to back philosophical statements at variance with the revealed Truth, or when evolution is held as if it were a matter of Faith, that the Church takes issue.
Pope Pius continues, however:
The faithful cannot embrace that opinion which maintains that either after Adam there existed on this earth true men who did not take their origin through natural generation from him as from the first parent of all, or that Adam represents a certain number of first parents. Now it is in no way apparent how such an opinion can be reconciled with that which the sources of revealed truth and the documents of the Teaching Authority of the Church propose with regard to original sin, which proceeds from a sin actually committed by an individual Adam and which, through generation, is passed on to all and is in everyone as his own.
In other words, if Adam and Eve were not a single pair of people, who sinned, and who were the ancestors of all subsequent humans, it would be possible to deny that all human beings are subject to original sin. Since this is a matter of Faith—something which cannot be denied without committing heresy—there appears to be no way to hold that Adam and Eve were not literal beings.
The Pope concludes by mandating that teachers of the Catholic faith not present such statements (e.g. that Adam was not a real person, but represents a number of early humans) as fact.
Note: To believe that there were two humans who were the ancestors of all subsequent humans is scientifically reasonable, as I understand things. That is, it appears that the statement found above in this question:
The theory of evolution ... implies—if we don't misinterpret the theory—that Adam and Eve never existed
may be false without contradicting what is known scientifically about humans; and the Catholic Church, as appears in this document, officially believes that it is false.