This is a wonderful 12-minute clip in which the full ballad is included, along with the beautiful song "They'll Never Make Her Cry," other musical interludes and some brief dialogue.
One of the things that made the song such an odd nominee is that it's not really performed as a song per se, in the movie. If you've never seen the film (and if not, then when you finish this, go get it from Netflix or wherever you rent your movies), the song is a true "ballad" that interspersed throughout the movie to help narrate the story. It's an impressive bit of film work interpolating everything together.
("The Ballad of Cat Ballou" was written by legendary songwriters Jerry Livingston and Mack David. David had eight Oscar nominations, and Livingston three. By the way, Mack David was the older brother of Hal, who was Burt Bacharach's longtime collaborator.)
Making the song additionally fun in the film was the unlikely pairing of the two balladeers, Nat King Cole and Stubby Kaye. Nat King Cole was a renowned popular singer with such huge hits as "Mona Lisa," "The Christmas Song" (Chestnuts roasting on an open fire...), "L-O-V-E," "Ramblin' Rose," and "Those Lazy-Hazy-Crazy Days of Summer," and many more, who was also the first black performer ever to to have his own national network TV series. Stubby Kaye came from a different world, he was an extremely successful Broadway supporting actor, whose most famous shows were probably Guys and Dolls (in which he introduced the song, "Sit Down, You're Rockin' the Boat") and Li'l Abner, which he recreated on film, doing the joyous "Jubilation T. Cornpone." Cole was smooth as silk, and Kaye was...well, showbiz, loud, stubby. So, it was bizarre that someone thought of putting them together. But it was inspired, because they're a joy.
(The film received five Oscar nominations, including Best Music Score, and Best Adapted Screenplay by Frank Pierson. And it's worth noting that in 2008, the American Film Institute named it #10 on the list of Best Westerns of all time, not shabby at all for a comedy. For that matter, AFI also ranked Cat Ballou as the #50 Funniest Movie of All Time. To repeat what I said earlier, If you haven't seen it yet, do yourself a favor.)
By the way, the video includes one of the great entrances in movie history. Catherine Ballou has written a letter to the legendary gunfighter, Kid Shelleen, hiring him to help protect her father from the town elders trying to steal his land. She excitedly waits for the arrival of the mean, rough, fabled gunslinger -- the adventurous hero of dime novels -- at the stagecoach. It's the first step on the road which helped win Lee Marvin the Best Actor Oscar.
For that matter, the movie also helped launch Jane Fonda's career. She was moving in that direction, but it'll be clear from this why Cat Ballou helped her along that way.