Hondas of that age seem to have problems with the ignition switches. It's a pretty simple fix. Remove some dash panels, pull out the old one, put in a new one. Shouldn't take more than an hour or so.

Easy way to verify this is to instead of letting off the key all the way, keep it pushed over toward start a little bit -- not so far over that the starter keeps turning, but as far over as you can get with it not running the starter. If the car stays on, there's your problem. This doesn't always work, so if it doesn't keep the car running, it doesn't necessarily mean that this isn't the problem, but if it does keep the car running, it's definitely the ignition switch.