And what, if anything, does the constitution say about secession?
The Constitution does not
specifically say one way or another, but the prevailing view has always been that it is not permitted. The Founding Fathers had a huge debate on the subject but eventually decided that in order for the Federal government to have any kind of actual power, the States could not just leave any time they happened to disagree. It was believed that any major controversy could cause the States to "go the way of Europe" and become a bunch of small countries and have constant petty wars.
In any case, the 1868 Supreme Court (the guys whose job it is to decide what is or is not "Constitutional") ruled that it is illegal for a State to secede from the Union, which has held up ever since.