One of the fun things about the sing-along finale is that, amidst the "sing-along gang", there would often be a celebrity hidden amongst them that the camera would pick up as it passed by. You had to pay attention, or you might miss it. And there is indeed a celebrity in this particular clip, and one who most people here should recognize very easily. (I won't say who, or when he appears...) Oddly, though, there's no need to pay close attention because, completely out of character, there's no "quick passing by" here, but they not only linger on the fellow for a long time, but actually several times. But he seems to be having a grand time about it all.
A couple of other small notes worth mentioning. It's not generally remembered, but the Tony Award-winning singer-actress Leslie Uggams got her start on Sing Along with Mitch, and she makes an appearance near the end of the medley. And also, one of the "gang" who also was a soloist on the show is Bob McGrath, who most people best know as 'Bob' on Sesame Street, which he played for a few decades. You'll first catch a glimpse of him at the 2:02 mark, and then he too shows up at the end of the medley..
I'm not quite sure what year this is from. The iMDB lists the show from 1961-1964. But a voiceover promo at the end is for the series, Mister Roberts, which aired in 1965-66. And the promo for a Stanley Cup hockey game suggests it would be 1966. It could be a repeat, of course, though several years later would be odd. Maybe it was in syndication, though? Moreover, to confuse things more, this is in black-and-white, and in its later years the show was in color. So...who knows? But it seems early-'60s.
[UPDATE: According to our friend Walter J. Podrazik, TV maven and curator of the Museum of Broadcast Communication, the broadcast of this was indeed 1966, on April 29. He writes that, "It returned as a spring/summer series April to September 1966 running Fridays 8:30 to 9:30 (Eastern Time). Thus with the helpful hockey info it was possible to place it exactly. (Game 3 was Thursday, then the program, Game 4 Sunday)." In other words, iMDB has it wrong that the show only ran until 1964. When in doubt, I always trust Wally.]
So, let me hear a melody...