As long as they can keep the slave population captive; you're going to have a situation where demographics will favour the slaves within a generation (in 1860, there are 39% slaves in the CSA, only 1,5% free blacks). 60% of the population holding 40% of the population captive will very quickly, horribly degenerate. If you thought sharecropping and Jim Crow were bad, things are going to suck, even harder.
Two of the states are majority slaves (South Carolina, Mississippi), Louisiana, Alabama, Florida, Georgia are almost so; of those four, Louisiana is also majority black. Besides Louisiana, the only majorly industrialized states, North Carolina and Virginia, are the only ones with a significant free black population. A generation later, the four other southern states might be majority black, and you end up with a demographic balance where the West and North of the CSA is still majority white, and wealthier, while the southeastern states are basically in a constant state of internal civil war if they don't give up the peculiar institution fast, and they just fought for the damn thing, and the CSA constitution guarantees the right to own slaves, even if said state decides to go abolitionist.
And ATL where Cuba and the Dominican republic somehow become CSA states just make the situation even worse, now the white plantocracy is absolutely, completely screwed in the mid to long term.
Oh and to make things bleaker; industrial slavery will probably make things worse as the white working class will potentially lack the means to form a significant middle class, there might be one but more on the scale of the latin american republics at the time, at best, with only 6% of the population owning slaves. You'll probably have to have a hefty dosage of racial propaganda to maintain the order and even then the more educated parts of the white lower classes will probably learn of, and in part turn to, fascism or communism by the 20th century. And of course dependency on cash crops with only limited industry will have to be turned around with an ever stagnating economy and an aristocratic mindset in the upper class going against it. Slavery can last, but will the CSA?
Edit: I still say into the 20th century, assuming they don't get foreign pressure. At least it will last longer than Brazil.