There wasn't much of a debate actually. Christianity essentially got their privilaged position over night: Diocletian persecuted the church with vigor, but when Constantine was elevated to Augustus in the west, the persecution not only ended immediately, but the church got a major status upgrade. A common misconception is that Constantine made Christianity the official "state religion" of the Roman Empire. This is not the case. Constantine made many changes to Roman politics showing clear preference to Christianity, but he still permitted paganism to continue. With regard to his preferences, he gave Christianity imperial patronage (i.e. churches would be built by state funds, Bibles were commissioned, and he sat in at all the church councils, where he played no small role in the decisions made), made sure all of the people he hired to work in his administration were Christians (Which 20th century christian scholar Charles Williams said 'took all the mediocre pagans and converted them to half-hearted Christians), and even changed Roman law so a person on trial could appeal their case to an ecclesiastical court. Despite all these things, Constantine did not formally convert and get baptized until he lay dying. Constantine is important because he was the first to start Christianity's rise to power. His son, Julian, would become emperor after a short civil war. He tried to bring paganism back, but he failed, both because of his poor political planning and Christianity's general betterness in the sense of doing good deeds, taking care of the poor, and just having a better reputation. Moreover, Julian began to bring back blood sacrifices, which at this time in Roman history, made many quite uncomfortable. Sacrifices had devolved at this time into either burning incense or flour or grain of some sort. Julian just did not represent paganism very well, but he certainly tried hard. Julian would meet an untimely death in the east fighting the Persians, taking a Persian spear...in the back...very suspicious (many believe that a Christian soldier of the Roman army was responsible, but there is no evidence. Most of the historians of the time were Christians, so they tended to view this as God's judgment visited on Julian by Persia). The official "end of the debate" came when Theodosius the Great made Christianity the official state religion, although paganism would still exist. Theodosius did a lot for Christianity. He was called "the great" for his zeal in the faith rather than his superb administration of the empire. This should not undermine his claim to excellence, however. He was a competent leader, good general, and was something more than mediocre, a rare trait for the late empire. Theodosius fought the western imperial duo, who were both pagans, in the Battle of the Frigidus, which was seen as the triumph of Christianity over paganism. On the first day of the battle, Theodosius was turned back by the pagan emperors, but the following day he renewed the attack and won, though only by a small margin. This did not stop the panegyric poets from styling the battle as God visiting vengeance on his foes, however. So, to sum up, the debate was more or less a transition, though not always a peaceful one. Paganism continued for at least a few hundred years after, but it did eventually die out.

Some sources, if you want to read more:

Eutropius-a pagan historian who very well shows Constantine's less noble side. He balances quite well the dynamic duo that is Eusebius and Lactantius. Eusebius, Ecclesiastical History-Eusebius is very propagandistic, but that does not necessarily completely invalidate him as a source. He still had incredible access to great primary sources. He was also a fine writer, trained in the classics. As you read him, make sure you take what he says with a grain of salt Lactantius, On the Manner in Which the Persecutors Died-Lactantius was also well educated and well trained in the classics. He is much more honest than Eusebius, while still having similar access to the emperor. Ammianus Marcellinus, The Later Roman Empire- Ammianus was a soldier who, although from the east, wrote in Latin, something which the Christian authors did not. He was a pagan, but he is willing to give credit to the Christian emperors, something Eutropius was not willing to do. You will find that he is a very balanced source, something quite rare in this time. I would also recommend the "Encyclopedia of Ancient Christianity" (IVP, 2014). Top-notch patristics scholars and ancient historians contribute to it, and it also lists excellent sources at the end of every article. If you are interested in the battle of the Frigidus, Arther Ferrill gives an excellent summary of the battle in "The Fall of the Roman Empire: The Military Explanation". Hope this answers your question!