I came across a website (Florida Folklife from the WPA Collections, 1937-1942) where you can listen to an early rendition of this song. They call it "Bellamena"

Here's the URL:

http://memory.loc.gov/cgi-bin/query/h?ammem/flwpabib:@field%28NUMBER+@band%28afcflwpa+3384a1%29%29

They are slow to get started (warming up to it I guess), but the tune and words are very recognizable. Seems the song may have originated in the Bahamas. Here's what the site says about the song:

Item Title: Bellamena

Performer: Rolle, Theodore (Tea Roll); piano

Created/Published: January 20, 1940

Notes

duration: 4 minutes, 28 seconds

A Bahamian folksong, commemorating the role of Caribbean vessels in "rum-running," or transporting alcohol, to the United States during Prohibition (1919-1933). The 18th amendment to the Constitution, known as the Volstead Act, or Prohibition Act, had outlawed the manufacture and sale of alcoholic beverages in the United States. Enterprising "rum-runners" transported alcohol manufactured from islands in the Caribbean to the U.S., repainting their boats between trips to avoid suspicion by the U.S. Coast Guard.

The singer is from Andros Island, Bahamas.

My favorite version of the song is on the "Heart of Oak" album. I'm looking for the music.

-Rand.