#Our Lady of Lourdes appears to St. Bernadette and tells her "I am the Immaculate Conception".
Joan Cruz's See How She Loves Us: 50 Approved Apparitions of Our Lady, § "Our Lady of Lourdes":
Members of the family and countless visitors had implored Bernadette several times to ask the identity of the Lady. Bernadette had done so, but the Lady had not responded. During the sixteenth apparition on March 25, 1858, the Feast of the Annunciation, Bernadette asked once more. This time the Lady responded in the Bigourdane dialect, “Que soy era Immaculada Conceptiou,” which means, “I am the Immaculate Conception.”
In the year 1854, four years before the start of the apparitions at Lourdes, Pope Pius IX had declared [in Ineffabilis Deus] the doctrine of the Immaculate Conception of the Virgin Mary to be an “article of faith” to be believed by all Catholics. The doctrine was neither understood nor discussed among persons of Bernadette’s station in life,* so it was reasonable that Bernadette did not understand the words spoken by the Lady.
*Her family was extremely poor, living in a barn with a window facing a dunghill.
St. Bernadette—who was 14 years old, unlettered, and failed her catechism exam—couldn't even pronounce "Immaculada Conceptiou" when she told others this was who the Lady said she was. However, priests certainly knew that the Immaculate Conception is the Mother of God. There is no way St. Bernadette could have fabricated this, which proves the miraculous nature of the Our Lady of Lourdes apparitions.
#Why she said "I am the Immaculate Conception" and not "I am immaculately conceived"
Fr. Karl Stehlin's Who Are You, O Immaculata? p. 114 (PDF p. 132) explains:
At Lourdes the Immaculate Virgin answered St. Bernadette: ‘I am the Immaculate Conception!’ With these words she clearly stated that she is not only immaculately conceived, but is the Immaculate Conception. In the same way, a thing that is white is something other than whiteness itself, and a perfect thing is something other than perfection itself. When God spoke about Himself, He said to Moses, ‘I am Who am,’ that is, it belongs to My essence that I always have my being from Myself, without beginning. In contrast, the Immaculate Virgin has her beginning in God, is a creation, is a conception. Nevertheless she is the Immaculate Conception.
She is, as St. Maximillian Kolbe put it, "the quasi-incarnation of the Holy Ghost." She and the Holy Ghost can both be properly called the Immaculate Conception (ibid. pp. 50-51, PDF pp. 68-69):
We can say likewise that she is the greatest, most excellent, purest Temple of the Holy Ghost. Mary herself corroborates this truth when she defines herself at Lourdes: “I am the Immaculate Conception” and thus assigns to herself the title that in the strict sense is an attribute of God (“I am...”) and is applied in particular to the Holy Ghost, who within the Trinity is the eternally perfect, “immaculate” conception of the Father and the Son.
If among creatures a bride takes the name of her husband by the fact that she belongs to him, unites herself with him, makes herself like unto him and together with him becomes the source of new life, how much more should the name of the Holy Spirit, "Immaculate Conception", be the name of her in whom He lives with a love which is fruitful in the entire supernatural economy?4
4. Final article of February 17, 1941, KR 212-213
(taken from this answer)