Actually this is very easy to repair and I have done it several times on bikes and other stuff.
You'll need a drill, very small drill bit, Allen key/hex wrench, small hammer and a punch.
1/. Wrap a cloth around the surrounding frame to protect it while doing the work.
2/. Using the punch place it dead center on the bottle cage Allen bolt / Hex bolt and using the hammer gently tap the punch to create an indentation. Repeat several times so that you get a nice indent. This is the guide for the drill bit.
3/. Now using the drill on a low setting and using a drill bit of 2mm or 3mm drill a whole about half way in to the broken bolt. Go slowly as you don't want to slip a gouge the frame.
4/. Now you have a whole select an Allen key/hex wrench that fits tightly into the whole that you drilled. You may need to use the hammer to tap the wrench into the hole. (Actually the tighter the fit the better)
5/. Now working slowly unscrew the bolt..... Hey presto you should now be able to replace the bolt.
If this fails you can either drill all the way through the Allen bolt and then when you apply torc when unscrewing bend the wrench at an angle so it catches on the bottom and top of the bolt or you may need to totally drill out the bolt. I normally do this in several stages, following the steps up to 3 and then using bigger drill bits until only a small fraction of the bolt remains, then using snip nose pliers either tearing the remaining shell of the bolt out. I recently had to do this to a chain ring bolt.
LBS job, looks like it being the bottom screw on the ST you might be squeezed for room to get a drill in there(if you have a drill). Pay the money and let the bike mechanic worry about damaging your frame. Did you over tighten it or was it stuck in there? They may want to know which cause if if was the former, Far Easts solution probably won't work. There is something called an "easy out" that also works if you can get the smaller drill bit in there. Good luck.
My solution is the 1st thing the LBS will try, followed by completely drilling out the bolt.
As the head of the screw has been snapped off and not striped there is no downward presure on the actualy bolt it's self so it should be very easy unless as others have pointed out the screw has bonded with the frame.
The bike is practically new, so there shouldn't be any salt or sweat that seeped in there. This an Yahoo auction item that I just won, and I still haven't taken delivery of the bike yet. Seems like the bottle cage broke off, and the LBS tried to remove the screw. While trying to remove the screw, they broke off the head of the screw.
I guess I will be taking this to a LBS to see if they can fix it. Are they boys at Asahi CBA any good, or should I try somewhere else? The guys at Asahi do a great job with my mamachari (all three of them), but I am not sure if they can handle a more complex bike.