Hmmm...
Well, as much as I'd love to see a TL where the CS becomes independent without a civil war..it's quite hard to pull off.
Anyway, here's a shot:
The way that I see it, if you get Abe Lincoln to die before he becomes the Republican golden boy and someone less daring(with similar amounts of popularity) replaces him and wins the presidency, we can have a good shot.
The CS secedes per OTL, as do the OTL 'border states' who attempted to secede (Maryland, Kentucky, etc...) and succeed unmolested (no unconstitutional arresting of legislatures).
The unfortunate union president, unwilling to face a Confederacy bolstered by the borderstate populations to levels similar to that of his own nation in a bloody war, acknowledges the CSA as a legitimate and sovereign nation. Britain, France, and Prussia follow shortly, offering alliances and trade aggreements.
Boom.
USA? Check.
CSA? Check.
No civil war (yet)? Check.
That's my two cents.