Huon

Thanks to Roger Pearse, we have the report by Philostorgius of the death of Constantine :

http://www.tertullian.org/fathers/philostorgius.htm
see Book 2 chapter IV of his history.

CHAP. 4.--Philostorgius asserts that Constantine was induced by the fraudulent artifices of his step-mother to put his son Crispus to death; 18 and afterwards, upon detecting her in the act of adultery with one of his Cursores, ordered the former to be suffocated in a hot bath. He adds, that long afterwards Constantine was poisoned by his brothers during his stay at Nicomedia, by way of atonement for the violent death of Crispus.


And also thanks to Photius, who reported what this dirty Arian Philostorgius wrote !

Huon

Another source :
http://www.roman-emperors.org/impindex.htm

Shortly after Easter (3 April) 337 Constantine began to feel ill. He traveled to Drepanum, now named Helenopolis in honor of his mother, where he prayed at the tomb of his mother's favorite saint, the martyr Lucian. From there he proceeded to the suburbs of Nicomedia, and there he was baptized, as both Eusebius and Jerome report; but only Jerome adds another significant fact: the baptism was performed by the Arian bishop Eusebius of Nicomedia.

A few weeks weeks later, on the day of Pentecost, 22 May, Constantine died at Nicomedia, still wearing the white robes of a Christian neophyte.