Do your brake lights work? (key on, key off?) They are on the same fuse, maybe there's something funny about the fuse or box. That or maybe there's not enough power for both... try running at about 4,000 rpm and press your horn, then horn and brakes!
No, really, I think you should check it that way because your alternator will be supplying power at that rpm. Maybe there's something screwy about how much power the circuit is getting. Just a thought. Try switching the fuse out, at least.
If all else fails, (and you don't have air bags) try removing the horn button from the wheel. It pulls straight off. Maybe you can see a problem that way. Maybe try it after dark, and see if there's a spark. Probably not, as this would blow the fuse...
Last resort, pull the steering wheel if you have to to get a look at what is going on in there. (my '65 MGB's horn blew every time I made a right turn! LOL)