Amendment 10 of the Constitution states that any right or privilege not exclusive to the Federal Government belongs to the states. One of the Confederacy's arguments was that secession was one of those rights. Of course, as noted above, the rest of the country has to agree. Since it didn't back then, that put the Confederacy itself in violation of Article Four, Section Three:
New States may be admitted by the Congress into this Union; but no new State shall be formed or erected within the Jurisdiction of any other State; nor any State be formed by the Junction of two or more States, or Parts of States, without the Consent of the Legislatures of the States concerned as well as of the Congress.
Of course, nobody really has to be asked. The Louisiana legislature could just vote to secede and see if they have better luck enforcing it today than the entire Confederacy had way back when.