In Andy Williams’ recording of this song, the instrument that carries the rhythm seems to be the contrabassoon. Is it? I’ve never heard of this instrument being used in popular music…
Good God, I was praying that Andy had Andified a different song with the same name.
Campers, if you are Brecht and Weill fans and value your sanity DO NOT CLICK THAT LINK!
When we heard that song as kids, my older brother sang one line as, “That old Bill Bowel movement…” 
As a longtime player and devotee of that instrument, it is with great regret that I say yes, it is a bass saxophone. Because that record achieves epic levels of suckitude.