This is a very early video of her work in the 1959 Broadway musical Redhead, which is also notable because not only did Bob Fosse serve as choreographer, but it was also his first directing job, at her insistence. She was coming off two big hits -- New Girl in Town and the huge Damn Yankees -- so she had that kind of pull.
Redhead had a very respectable run of 452 performances, a little over a year. It won six Tony Awards, including Gwen Verdon winning for Best Actress, Richard Kiley (who a few years later would start in Man of La Mancha) who won as Best Actor, and Fosse won for choreography (though he didn't get a directing nomination), and the show itself won as Best Musical. And the show had a good pedigree on the production end. The lyrics were by the legendary Dorothy Fields -- who wrote such shows as Sweet Charity, A Tree Goes in Brooklyn, three musicals with Cole Porter, won an Oscar for the song "The Way You Look Tonight," and co-wrote the musical Annie Get Your Gun). And the music was by Albert Hague, who is probably better known to people today for playing the music teacher, Mr. Shorofsky, in the movie and then TV series Fame. Moreover, one of the show's book writers (along with Dorothy Fields and her brother Herbert) was Sidney Sheldon, who would later go on to great success as a novelist, including The Other Side of Midnight and Bloodline, and created the TV series I Dream of Jeannie and Hart to Hart.)
But this is about Gwen Verdon, and so here she is. By the way, it's also sort of amusing that has a bowler hat in the number, which somewhat became a Fosse trademark. As far as I can tell, the appearance is from the Dinah Shore Chevy Show.