1. The Confederacy fights a war of conquest aimed against Spanish Cuba. The Confederacy then sues for peace when the Spanish mount a successful counterinvasion of Florida while the Union looks on, caught between a desire to punish a Spain violating the Monroe Doctrine and wanting to see the Confederacy humbled. Extra points if you can justify a Mexican invasion of south Texas.
2. Rather than see the Union and the Confederacy drawn into the European alliance system on opposing sides, simply have the Confederacy--heavily dependent on Anglo-French capital--absorbed into the Allies' ranks. Years of bloody trench warfare later, the best of the white Confederacy's young men lie, shattered and rotting, in northwestern Europe's damp soil. The intervention of a neutral prosperous Union on the side of the Allies late in the war, after the Confederacy is bled white, is optional.
3. Because of the adoption in the Confederate constitution of a variety of economic policies sure to doom the Confederate economy to long-term collapse, large-scale emigration to the rich Anglophone Union is likely an inevitability. Because a racial bias against Afro-Confederates in Union immigration policy is likely, it will be Confederate whites who flee north. The result: A Confederacy that, a century after its foundation, is mostly Afro-Confederate in population. Paging alt-Mandela . . .
It never fails to amaze me, or irritate me, to read alternate histories where the Confederacy becomes a major world power. Consider: Are agrarian economies which keep more than a third of their population as slaves, which are so hostile to productive state interventions that they ban these in its constitution (see Sections VIII and IX in Article I), and which are run by an exclusive aristocracy obsessed by literature-inspired theories of racial purity destined for prosperity? For all of its late-twentieth-century problems, at least the Southern Hemisphere didn't have its potential investors discouraged by masses outraged by the persistence of chattel slavery. And since the matter of slavery was the issue that started the American Civil War, you can speculate with some reliability that the Confederacy would have kept its slaves for as long as it could.
The myth of a successful Confederacy is strikingly similar to the myth of an efficient Nazi Germany, inasmuch as protagonists of these two myths ignore the historical realities that, in fact, these two states were not only illegitimate but fundamentally flawed. Nazi Germany was an inefficient state vialb eonly because of booty from its conquests: Had Czechoslovakia chosen to fight in 1938, for instance, not only would it have bloodied the German army and deprived Germany of vital military stores, but the inability to loot an intact Czechoslovakia would have provoked economic collapse in Germany. Similarly, everything that we know about the policies favoured by the Confederacy's erstwhile elite suggests that in a globalizing late-nineteenth-century world, it would have found itself increasingly out of step not only with a global economy that required participant states to be run competently, but with a global civil society that increasingly if imperfectly cared about oppression abroad.
UPDATE (6:39 PM) : Browsing in the soc.history.what-if archives on Google Groups, I find this post by Raymond Speer going into more detail on the potential scope for extreme Confederate dysfunction.