First, then, you have to get rid of the OTL Civil War. I'm assuming, without doing a search, that there are hundreds of discussions on that scattered around this forum. Take your pick of the best one. How about the Republican Party never really comes to a complete agreement, and remains as fractured as the Democratic party in the 1860 election, leading to a long series of very weak presidents.
This leads to no emancipation proclamation. Arguably, slavery is still abolished gradually by around 1900, perhaps through government subsidies to slaveowners who voluntarily free their slaves. However, it continues as a social institution under different terms (African Americans are now "paid", and are free to find other jobs, but social pressures make this all but impossible).
Various riots and rebellions, are used by Southern state governments to justify military enforcement of the status quo. The Federal government simply ignores and tolerates all of this as a side effect of keeping the nation stable.
The U.S. has not industrialized as much as it did in OTL, and is essentially a second-world, if not third-world nation. It remained neutral in both of the *World Wars, and its few expansionist policies generally failed. Meanwhile, Mexico, which fought a war with Spain in the late 1890's which gained it the Phillippines, Cuba and Puerto Rico, is a highly stable, progressive, democratic republic and one of the wealthiest countries on Earth. Due to its dependence on cheap labor and agricultural products arriving from north of the border, it has an interest in keeping the U.S. from destabilizing and essentially supports the status quo.
Finally, political winds change in the late 1950's, and when yet another rebellion occurs in 1960, states throughout the North and West agree to send their National Guard units in to settle the issue for good. Southern States, angry about the interference in their affairs, fight back and try to impeach the administration which supports this, but lack the votes to succeed. They sue, but the Supreme Court is full of northern sympathizers and throws it out of court. Civil War breaks out in 1961 when the National Guards of the Southern States begin taking back areas occupied by the North. There's no secession here: instead, the Southern States initial, primary goal is to simply capture Washington, D.C. and force what they feel is an illegal government to abdicate and replace it with a truly elected government.
Much of the rest of the world is currently locked in a scuffle with Communists in Southern Europe and Eastern and Central Asia. Mexico and Canada and Great Britain, who have important interests in having a stable U.S.A., intervene, but to what end...