We are speaking of the Immaculate Conception. The Immaculate Conception is Joachim and Anne's (Mary's parents) conception of Mary, not Mary's conception of Jesus. While Mary's conception is celebrated in the Catholic Church, Jesus' conception is not. The Annunciation of the Angel's message to Mary about his conception and birth are celebrated (March 25 of each year). Mary's own Immaculate Conception is celebrated on December 8.
The virgin birth, of course, is celebrated on December 25, nine months after the Annunciation. Tradition has it Jesus died on the day he was conceived (March 25). No, that date has never been proven.