I'm cribbing this from my pal Bob Elisberg, who found it online. It's Rex Harrison performing "I've Grown Accustomed to Her Face" from My Fair Lady…twice. The first part is from an appearance he made in 1981 promoting the "farewell" tour he was then on with a revival of the show. I saw him at the Pantages in Hollywood then and…well, it was great to be able to say "I saw Rex Harrison do My Fair Lady" but neither he nor that production really sparkled as the original must have. He forgot a lot of lines, including part of the lyric in this particular song, and there was a certain lack of energy throughout the whole show.
Anyway, the most interesting part of this video is the second rendition of the song, which is from the original 1957 production. Today, it's not difficult for someone to take a camera into a theater, even unauthorized, and capture a performance. Back then, this must have been a bulky 16mm movie camera using existing light and I'd be fascinated to know who did it and how. (Actually, I think I know who did it and I'll verify that next time I run into the guy…)
One interesting point about this song. Harrison was an unusual musical comedy performer. He was not trained to be one and he did it with his own approach, which was just to start singing and let the orchestra follow him, instead of the other way around. As a result, it is said he never did any number exactly the same way as the night before. When they made the movie, its makers found it didn't work to have him pre-record his songs and then lip-sync in front of the cameras so they worked out a way he could sing them live on the set…and apparently, each take was quite different in both phrasing and physical action. Here's how he did it one time in '81 and one time before that in '57…