In thinking over the title One Desire I've concluded it refers to the One Desire that Anne Baxter and Julie Adams have for Rock Hudson. Each wants the whole Rock and not just a piece of it.
Baxter and Hudson play a pair of saloon denizens. She's a house madam and he's a gambler by trade. Both decide they can do much better, especially after Hudson's little brother Barry Curtis arrives with nowhere else to go. Later on the two take in Natalie Wood an orphan girl whose father is killed in a mining accident.
This nice jerry built family is interrupted when Adams who is the spoiled daughter of a Senator and bank president Carl Benton Reid sets her cap for Hudson. She connives to get the kids taken from Baxter as an unfit parent and Hudson being no match for her wiles marries her and Baxter goes back to her old trade.
The surprise here is Julie Adams who usually plays nice women. This was a bit of offbeat casting for her, but she pulls it off. She's as manipulative as Eve Harrington.
An interesting tale, the women are fine, I could not quite grasp Hudson's character as so naive especially with his background as a gambler. You're supposed to learn to read people in that trade.