Saint Hippolytus (A.D. 170-240) wrote in passing that the birth of Christ occurred on December 25:
“The First Advent of our Lord in the flesh occurred when He was born in Bethlehem, was December 25th, a Wednesday, while Augustus was in his forty-second year, which is five thousand and five hundred years from Adam. He suffered in the thirty-third year, (April 3, 33 A.D.,) Friday, the eighteenth year of Tiberius Caesar, while Rufus and Roubellion were Consuls.”
313 A.D. The Early Christians built a Church over the site of where Jesus was born, Constantine was by this time a Christian.
379 A.D. The preaching of Saints such as St. Gregory Nazianzus spoke of Christmas continuing the tradition of December 25.
Saint Augustine confirms the tradition of March 25 as the Messianic conception and December 25 as His birth:
For Christ is believed to have been conceived on the 25th of March, upon which day also he suffered; so the womb of the Virgin, in which he was conceived, where no one of mortals was begotten, corresponds to the new grave in which he was buried, wherein was never man laid, neither before him nor since. But he was born, according to tradition, upon December the 25th