Well, it's complicated. Constantine patronized Christianity, so in some sense it took the place of the old religion as the "official" religion: just more like "D'Angelo's: The official sandwich of the Boston Red Sox" than "the one everybody had follow". Pagan cults, on the other hand, weren't banned until Theodosius in about 390, so one could take that as the point at which Christianity became the "official" religion in the latter sense.
But get a second opinion from sovay.
--The Professor in question