I thought I'd post the trio's recording of the song, but first wanted to mention why that review of up-and-coming artists is considered "famous."
For starters, consider the cast of then-unknowns. Paul Lynde, Alice Ghostley (who later would co-star with Lynde on the TV series, Bewitched), Eartha Kitt, Carol Lawrence (who created the role of 'Maria' in West Side Story), Robert Clary (who came to fame as 'LeBeau' on Hogan's Heroes), Ronny Graham (a writer and actor whose career often intertwined with Mel Brooks) and others. But further, among the people who wrote the show were Mel Brooks and Sheldon Harnick, who later went on to write such musicals as Fiddler on the Roof.
That's not too shabby a collection of unknowns. All in all, it ran for an impressive 365 performances.
As I sought out the recording by the Chad Mitchell Trio, however, I made a discovery.
The song, with the original Broadway cast, actually exists on video!
It turns out that two years after appearing in New York they made a film version of the show, with a slight plot added, and called it New Faces. The number is quite a bit different than the popular recording, since you get the sketch built in with a story setting everything up first, and here it's known as "Fall River Hoedown." (That's Ronny Graham who appears at 2:08, which is when the more-familiar part song starts up.)
So, rather than the Chad Mitchell Trio, to my great surprise...here is original. The whole thing!