Maybe this should move to a forum since this is a Nancy Sinatra record, but, much as I LOVE Teresa, I don't think that video is live. There are places where her lip movement doesn't feel right. The biggest thing to watch for is how the artist is breathing.
Having worked sound, the biggest problem is technical. How the heck do you mic somebody when they're moving around like that?
Trainman, I agree that's the best way. Add to that all the location settings in Nancy's.
But Teresa Brewer said she always sang live on TV shows (except when they had no orchestra). The way they often are moving around it's hard to think they are actually singing at the same time. See her on Shanana on Youtube, and what do you think?
That was pretty much standard procedure or any musical TV show. When I was in college learning broadcasting we'd make an audio tape of all the musical numbers in the studio, with the cameras on and a full crew, so all the ambient background noise etc. would match, then when we taped the actual broadcast, we'd dub in that tape, playing over the studio monitors with mics off so the talent could sing along, making it look natural.
She did a lot of lip-syncing on the show, so many of these are the recorded hit versions, like Friday's Child.
I also see that the TV show is on a 2000 DVD.