Countdown to Christmas: The Christmas Song (1957-63)

Nat "King" Cole, I think, though I cannot be sure, recorded Mel Torme's "The Christmas Song" with a jazz trio in 1946, then reissued it as a solo track 15 years later, and it is the solo version that everyone remembers.

Edit, 12/24/25: Had to change the video. Here's Nat on The Danny Kaye Show in 1963: