Wow - I'm so glad that I finally found this!!!! "This" being a song sung by Carolyn Hester called "Swing and Turn Jubilee". It was one of the first songs that I really liked when I started getting seriously into folk music up in Oregon, and it's taken quite a while to track it down again!!
It's a great song on its own - a lovely old square dance from who-knows-when - but this version has a neat little story to it:
Carolyn Hester was signed to a record contract with Columbia by John Hammond in 1960. He expected her to be the next big thing in the folk music revival that was happening at the time. Unfortunately, her first two albums didn't do so hot, so Hammond tried to get her a really good band for her third try.
If you listen to the playing on this song, it's just amazing!!! When I first heard it, I was particularly struck by the great harmonica breaks in between the verses. looking into it a little, I finally found out who played on it: A little known folk singer named Bob Dylan, recently arrived in New York and playing on his first album!!!
It may just be my imagination, but whenever I listen to it, I can just feel his excitement at being in the studio for the first time!! And by the way, on bass is Bill Lee, father of Spike Lee!!! It's just
such a great performance!!
Here's the band during the recording:

The album still didn't do so great, and Carolyn Hester never made it as big as her friend and rival Joan Baez, but I'll always have a soft spot for this song!!
Here it is from Last.fm:
http://www.last.fm/music/Carolyn+Hester/_/Swing+And+Turn+JubileeLive and learn!!