No they won’t. The North will oppose such a conquest, and no insignificant parts of the South would oppose it as well, as they didn’t want to bring a Mexicans into the country:The South conquers Mexico and asks for concessions on labor and gets them claiming Mexico is a territory.
“To incorporate Mexico, would be the very first instance of the kind of incorporating an Indian race; for more than half of the Mexicans are Indians, and the other is composed chiefly of mixed tribes. I protest against such a union as that! Ours, sir, is the Government of a white race.”
-John C. Calhoun
There might be a path for the South to force a conquest of Cuba, but even under Buchanan and Pierce such plans were not well supported.
Anyhoo, to the question: slavery lasts at minimum another 20 years. How long depends on how long the South can bluster and threaten to keep things paralyzed. Eventually some kind of gradual emancipation plan is agreed to simply because the political power of the North has grown too large to resist, and secession is far too late.