I think it was also done by Paul Simon, but I have no idea who originally wrote the lyrics.
It's actually a hit from the 1950s. My dad recognized it when I played TMBG! The 'Four Lads' did it back then, and it was a top ten hit in 1953. But they might not have been the first, because other bands might have gotten to the music for other tunes. I know when I looked at the album notes the copyright was from the 1920s.
The author, Jimmy Kennedy also wrote "South Of The Border (Down Mexico Way)", and co-wrote one of my favorite Bing Crosby tunes, "How Can You Buy Killarney?"
Lyrics by Jimmy Kennedy
Music by Nat Simon