Recently I made an exquisite trade through Craig's list - 5 beat up student violins and a student model trumpet, in exchange for a 90 year old Alto Horn. Since acquiring it I've played it a whole lot.
The problem was, the mouthpiece that came with it was for a mellophone, which has a shaft size that's too small for the alto horn. I set about increasing the shaft size instead of holding it together with scotch tape.
First I covered the mouthpiece shaft with a liberal amount of JB-WELD. Initially I thought it would be possible to sand it down by hand to a cylindrical shape, but this proved to be difficult.
My next idea was to sand while it was turning, as on a lathe. This required finding just the right size bolt and nut to secure the mouthpiece, but once done I was able to mount it in my hand drill. which I then mounted in my bench vise.
Then it was just a matter of sanding it down while the drill ran. I'm quite happy with the results, and glad I didn't have to go out and buy a new mouthpiece.