In short, since the title of the Articles of Confederation included the words "Perpetual Union", and the preamble of the Constitution includes "to form a more perfect Union", therefore it must mean (and in fact be inherently obvious) that individual member states shall not leave the union - even if the topic is not addressed in "meat" of either document.
Followon Q: If it was widely understood, why did all the states agree to that? It seems a stretch to go from the lower-case "u" united States of America (ok, we'll band together as it's the best way to fight the war and get International help) as implied in the Declaration of Independence to "The United States of America" in just a few years, what changed?
Earliest I can tell Texas v. White cements this as a Supeme Court ruling in 1869.
A bit late, but this comment by /u/turtleeatingalderman goes into the debate over the legalities of seccession.
Additional context: I'm American and am supposed to lead an International team on a several temporary assignements(Industrial facility startups) throughout the Southern US in the coming few years. Americans on the team with be me (Northern engineer), and several technicians from the South. International expats will come from Belgium, Spain, and/or Morocco and will also be technicians. I'm working on the "Welcome to the USA!" packages that are used to brief the whole team.
Among the things I have to discuss are state-to-state differences (within the US), tax implications, legal implications, etc...I'd like to tie that back to the Constitution and specifically why it is the way it is. I'm hoping to tie in a theme of "These regions were always different, and they structured the governement accordingly" which can be fed forward into discussions of elections, taxes, etc.. that always come up during deployments (with this being the first deployement we've done in anyone's home country).