Mance Lipscomb first recorded for Chris Strachwitz (Arhoolie Records) in 1960. Among the first songs he recorded, on a portable recorder set up in his home in Navasota, Texas, was Sugar Babe. This song, and the others he recorded in that session, were all released on two Arhoolie LPs later that same year.

As far as I could gather from a quick look at Tom Rush's website, he didn't record (at least not studio albums) until 1962.

Whether Rush learned his version from Lipscomb or not, I don't know, but it would appear to me that Lipscomb was first to record it. Although, David, you still might be right about Rush recording his version before Lipscomb ever went into a studio!

Brian