From Alan Lomax's "The Folk Songs of North America", Garden City, NY, 1960. p. 280:   
The "Rising Sun" occurs as the name of a bawdy house in two other traditional songs, both British in origin.... The melody can be linked with one setting of "Lord Barnard and Little Musgrove"... and with other old traditional tunes.'
    Yet this song is, as far as I know, unique. I took it down in 1937 from the singing of a thin, pretty, yellow-headed miner's daughter [Georgia Turner] in Middlesborough, Kentucky, subsequently adapting it to the form that was popularized by Josh White....
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Since Dave R. was not *singing* the song, and only said that the *tune* was old, I think he's in the clear... altho 700 years old is probably a bit of an exaggeration...