In 1861, big shots from the newly formed Confederate States of America met in Montgomery to draft a provisional constitution for their new country. Only sight changes were made between their provisional draft and the final one.
Not many changes were made to the original United States constitution - Jefferson Davis gave a speech saying that the new constitution was what the founding fathers intended, and differed only in that it made their original intention more explicit (in a very good example of the still-common fallacy of thinking that the original founding fathers all agreed on anything).
Of particular note is the addition of a line to the pre-amble which invokes the aid of "Almighty God," a concept left out of the United States constitution altogether. Confederate archives that were captured by the Union in 1865 give an interesting behind-the-scenes look at how this was added.
After some debate about whether to call the new country The Confederate States of America or The Republic of Washington, a motion was made to ad "invoking the aid of Almighty God" into the pre-amble. One man objected and wanted the line removed. Another wanted it changed to "invoking the aid of almighty God, who is the God of the Bible and the rightful source of all power and government." But, since this might imply that Christianity was the official religion, Judah P. Benjamin, the most notable Jewish confederate big shot, objected. The short "almighty God" line was kept; the longer one and more explicit one was left out.
There was also an attempt by either Thomas Reade Rootes Cobb, the be-mulleted gent above, or his cousin Thomas Howell Cobb, the guy below who looks like a larger, fresh-from-the-fight clone of Stephan Douglas (sources aren't clear about which Cobb it was), to add a line stating "No man shall be compelled to do civil duty on Sunday."
It was quickly rejected, so whichever Cobb it was tried to insert a law about at least banning the delivery of mail on Sunday. A guy from Louisiana said that the people of Louisiana believed that people could worship God any day they wanted, so Louisiana should be exempt from that law. Texas wanted out too, and in the end the whole line was removed.
Interestingly, the provisional authors voted down a rule saying that all new states joining the Confederacy had to allow slavery. However, since the final draft very explicitly protected the right to own slaves, such a rule would have been unnecessary. A non-slave state could have joined, but it would have immediately become a slave state.
For all the talk you hear about "State's Rights" being a major cause, there's precious little of it in the CSA constitution, other than some lip service to the states acting in their "sovereign character." That's the impression one gets from reading accounts of these conventions and meetings - they would talk about state's rights now and then, but there was only one that really loomed large in their mind. Once people got started talking about slavery and their frustration with northern attempts to end it, it was hard to get them to shut up.