There is no one prophecy exactly.
An early Welsh poem known as the “Song of the graves” gives the sites of the graves of many Welsh heroes, but then has the line:
“A mystery to the world, the grave of Arthur.”
See http://www.maryjones.us/ctexts/bbc19.html . The line is often translated somewhat differently, but the meaning always comes down to the location of Arthur’s grave is unknown.
In William of Malmesbury’s “Chronicle of the Kings of England”, Williams states:
“The sepulcher of Arthur is no where to be seen, whence ancient ballads fable that he is still to come.''
See http://www.fordham.edu/halsall/source/malmsbury-chronicle1.html .
In Geoffrey of Monmouth’s 12th century “Historia Regum Britanniae”, Geoffrey only tells that Arthur, mortally wounded in his last battle, is taken away to the mysterious island of Avalon to be healed of his wounds. And Geoffrey makes Merlin prophecy about Arthur that “his end shall be doubtful”. See http://www.lib.rochester.edu/Camelot/geofhkb.htm , Book VII, chapter III, and Book XI, chapter II.
Search on “Morgen” in http://www.sacred-texts.com/neu/eng/vm/vmeng.htm for a later writing by Geoffrey which tells Arthur’s fate in more detail.
See http://en.wikipedia.org/wiki/King_Arthur%27s_messianic_return and http://www.arthuriana.co.uk/n&q/return.htm for further medieval mentions. There is no single prophesy, just a common belief that Arthur is still alive.
There are a number of different beliefs, but none of these texts mention “reincarnation''.