Early Mashup

There are too many complications to claim that any one song is the "first" mashup. However, "Mission Impossible / Norwegian Wood" by the Alan Copeland Singers is one of the earliest that I know of. I came across this through Kembrew McLeod's article "Confessions of an Intellectual (Property)" as well as a comment posted on GYBO (Get You Bootleg On) by Timbearland. 

The song was arranged by Alan Copeland. The vocals were recorded, apparently, by The Alan Copeland Singers. I do not know anything more about the production of this track. 

Alan Copeland won the "Best Contemporary Pop Performance, Chorus" Grammy Award in 1968 for "Mission Impossible / Norwegian Wood." You can download this excerpt from the track at Mr. Copeland's website.

While I can't be sure, it sounds like both the vocal and instrumental sections of the track have been spliced to fit together (as opposed to being performed live or recorded live). The form of the track (A vs. B) and the method of construction (or my best guess at it) both fit perfectly with contemporary ideas about what makes a mashup.