Actually that is not correct.
The deal was made during the war between Constantine & Maxentius, circa 312. Constantine was seeking general support throughout the empire for assuming sole imperium. The early christian church offered him their political support in exchange for legal recognition & support of christianity throughout the empire. This was later cast by the church in the light of a "mystic vision" on the part of Constantine for purposes of spin & justification.
Constantine, an accomplished and very ambitious general, was also notorious as a manipulator & opportunist, skills at which he excelled. He was principally concerned with social stability in the empire, hence his desire to unify religous observances, both pagan as well as christian, as a means of minimizing areas of dissent.
He defeated Maxentius in 314 and subsequently legitimized christianity as a religion (Edict of Milan, circa 314).
His struggles for sole control of the empire went on for a decade. He finally achieved sole power as the Roman emperor in 324.
The Council at Nicaea occurred the following year, 325, after Constantine's accession to sole power. It constituted a restructuring of the church in order to take maximum advantage of their newly expanded political influence. Christianity now became the dominant officially recognized religion in the empire, although pagans were still officially tolerated by the government. The christian church became increasingly involved in the actual administration of government and used its influence to suppress others. Constantine supported this as he wanted to concentrate state power and eliminate potential sources of internal dissension.
Constantine died in 337. Christianity's OFFICIAL adoption as the sole legitimate state religion actually dates from after his reign.
His son and successor, Constans, subsequently outlawed the practice of pagan religions in 341, effectively making christianity the only legitimate religion in the state.
Julian, aka The Philosopher, aka The Apostate, Emperor of Rome, attempted to reverse the expansion of the christian church's state power by once more legitimizing non-christian religions in 361 and once more extending the practice of religious toleration. His reign was brief. He died suddenly and his attempted reforms were lost.
However, given all of the above, OFFICIALLY christianity became the sole state religion of the Roman Empire in the year 380. The fat lady sang in the form of a decree entitled De Fide Catolica by Emperor Theodosius I. By then the christian church had been dominating the administration of government for about 40 years. Theodosius not only specifically outlawed all forms of non-christian religion, he also outlawed all unorthodox or heretical christian practices.
Summary:
A. Constantine did his deal with the christian church circa 312.
B. First Council of Nicaea held in 325 to consolidate official christian doctrine as a part of Constantine's policy for enhancing political stability within the empire.
C. Christianity effectively made the state religion in 341 by Constantine's successors.
D. Christianity officially the state religion 380 by Theodosius I.
Mark A. Baker