It has always been my personal feeling that it *might* have died on its own, but no earlier than the 1910s.
The reason I claim this statement is because look at relations during the Mexican War era. John C. Calhoun in a defence of slavery called it a "positive good" in 1837, and many claimed that the enslaved African American was no worse off than the "enslaved" lower class laborer in the North. If anything, the slave had food and shelter, something a Northern factory worker was not assured of, or so the argument went.
For slavery to thrive it needed to expand. The frontier wasn't closed until 1890, so slavery could've had the chance to expand up until then. Once chocked off, it would have died after that, meaning that *maybe* slavery would have been eliminated by the time World War I historically occured.
If Southerners truly believed that slavery would die out, then why the violent reaction to the Wilmot Proviso of 1846, which would prevent the introduction of slavery into what would later become the Mexican Cession? Representative William Wick even tried to suggest simply extending the Missouri Compromise line which would theoretically have allowed slavery in the southern portions of the Mexican Cession, but this too was voted down.
The solution was the compromise of 1850 and popular sovereignty which would allow residents to choose to be a free state or slave state. Sounds good, but both slaveowners and abolitionists flooded the various regions (ala Bleeding Kansas) to fight over wether it would be free or, as it turned out, slave.
Because of the divisiveness of the issue and the massive amount of support the expansion of slavery had, it has long been my personal feeling that as of 1860 when secession began, there was no end to slavery in sight. However, as I said, as many of the slaveowning aristocracy claimed that slavery had to grow in order to work, it had to have ended rather quickly (my personal guess is 20-30 years) after the closing of the frontier in 1890. There would have been no place for it to expand to, unless of course the United States invaded Canada or Mexico.
Formerly of Company A, Second Colorado Infantry
"The boys rushed in, waist deep, with a yell that sounded like the shout of a thousand bull whackers." - Captain George West, Second Colorado Volunteer Inf., 1863, on the First Battle of Cabin Creek.