Before you begin, make sure the guitar isn't worth more than a few hundred bucks. If it is an expensive instrument, have it repaired properly by a professional. However, if it's a cheap guitar with plywood plates, you could probably by a new one for less than the cost of this repair, so the risk is small.
Other instructables about repairing non-electric guitars:
http://www.instructables.com/id/Guitar-Head-Repair/ (with style)
Step 1: Diagnose
In the case of my guitar, I found that the previous owner had strung it up with the wrong kind of strings. As you can see from the images below, the guitar was meant for nylon strings. There are structural differences between the two kinds of guitars. The differences are so great that you can feel it simply by lifting the guitar---the steel-string guitar weighs 2-3 times more than a classical guitar. All of that mass is due to the bracing and other measures taken to strengthen a steel-string guitar so that it can resist the pull of the taut strings. When my classical was strung up with steel strings, the neck broke off cleanly. Clearly, fixing the guitar won't solve the problem unless I use nylon strings in the future.