This first is that it comes from a black&white episode of The Tonight Show with Johnny Carson, in 1965. So many of those recordings are lost. In fact, this is from the days when Skitch Henderson was the band leader. (You hear him referred to.)
Also, Robert Goulet sings a song from a show by Alan Jay Lerner that hasn't opened yet, On a Clear Day You Can See Forever. (Not written with Frederick Loewe, who had retired by that point, but with composer Burton Lane, who had written Finian's Rainbow.) In some ways, the performance looks impromptu, since he says on the couch and sips water ahead of time to wet his throat. He's prompted by this then-wife Carol Lawrence, who was the original Maria in West Side Story. But since the band plays a lead-in and arranged version, I suspect there was a rehearsal, though it may have been left uncertain whether he would sing.
But the biggest rarity may be that Goulet gets Johnny Carson to sing a duet with him. Carson rarely sang, and then only a few bars, but this is a full number. It doesn't appear prepared for several reason, including that it takes him a long while to join in, and he seems awkward -- though he does a perfectly nice job.
Interest to note too that although the clip includes Goulet's first wife, it was posted here by his later (and last) wife, Vera. Hat's off to her.
(I got to meet Vera Goulet when I was the unit publicist on Naked Gun 2-1/2, and Robert Goulet co-starred as the bad guy. He seemed a nice enough guy, a little bit into himself, but personable enough, with a good sense of humor about himself. I think I've told a few stories about that previously. A good result of all that is that I was able to get him to sign a couple of cast albums to me -- one for Camelot, and the other for a moderate show by Kander & Ebb, The Happy Time, for which he won the Tony Award as best actor. I remember when he saw the album, his eyes popped and he said, "Good Lord, I haven't seen this album in ages.")