No. The Muppet Christmas Special where they all go to Fozzy's Ma's house has the Sesame Street gang singing it as they arrive. The words are:

Carolling, carolling, now we go,
Christmas bells are ringing,
Carolling, carolling through the snow,
Christmas bells are ringing.
Joyous voices sweet and clear,
Sing the sad of heart to cheer,
Ding, dong, ding, dong,
Christmas bells are ringing.

Elizabeth likes the "ding, dong, ding, dong" part. :)