This has to be one of the most frustrating things that can happen.
Check this out:

As you can see this bolt broke off with just enough left to get a good grip on.
Here is some different ways that you can remove a broken bolt depending on how bad it is broken.
The first thing I do is apply heat directly to the bolt.

Then, grab the bolt with some vice grips while it is still hot and try to unscrew the bolt.

This bolt is really stuck.
So, I took a nut that is larger then the bolt diameter.
The reason I use a larger nut is so that I can get some good penetration as I am a very poor welder

Place the nut over the broken bolt.

Weld the nut to the bolt.
Pay close attention so that you get good penetration.

As bad of a welder as I am, this weld turned out very good (For me).

Next, put a wrench on the bolt and attempt to unscrew it.

As I tried to unscrew the bolt I snapped the shaft right off.
So, I took a grinder and flattened the bolt off to get a good clean surface.
Sometimes the bolt will break deeper into the threaded area and then your stuck working with what you have.

Then I take a center punch and put a nice little dimple for the drill bit to get started in.
Try to get the dimple as close to the center as possible.

As you can see, I was a little off with the center punch.
It was still close enough to drill out the center of the bolt.

Then. I apply heat again.

Insert the Easy Out (This is my least favorite easy out. My good one was broke)/
Then try to back the bolt out.

The broken bolt came out and I couldn't be happier.
The threads were saved.
I did run a tap to the bottom just to be sure that there was no messed up threads.


All done.
If you have any questions or suggestions, feel free to post or email.
SB