I'm 5'10" and ride a 17.5" Trek 7.3 FX, which just fits me, in the sense that I've got the seat post up as far as it can go (it can probably go 8 inches lower). I probably could have done with a 20" frame, but the 17.5" frame does work for me. Like the Trek 7100 posted about above, the FX series also comes in 15" frames, and I'd imagine that a 15" frame would accommodate someone quite a bit smaller than me. You'd need to find a second-hand one to stay in your budget, but I think you'd struggle to find a decent new bike within your budget.
You could try a local bike shop to see if a 15" frame fits first, and then start hunting around for a second hand machine.
Last edited by fairymuff; 12-01-11 at 04:07 PM.