The (almost) complete appearance of the Carpenters on The Andy Williams Show transmitted on February 13th 1971.
I say 'almost' complete because they also performed We've Only Just Begun on the show, but as far as I'm aware, I've never seen it anywhere online, ever!
A huge thanks to Mr. Cuyler Otsuka for most of the clips used in this video.
You can visit his YouTube channel here: https://www.youtube.com/channel/UCJSYda1mJ-ceZUFd7-9Phyw