After it starts it will run fine for a bit then stops suddenly. Does not run down as it would with a fuel problem. Only happens when it is hot. Then it is impossible to start.

The most likely cause is the ignition module failing once it gets hot. While not common it is not unheard of either. I suggest you replace the spark plug first and try it (<$5). If no better try this cold just to get a feel for it, then as soon as it cuts out try it again.
Ignition Coil Test by SmallEng.com
http://www.youtube.com/watch?v=L7WNBDRG4C4
You may need to use a jumper wire between the threaded part of the plug and the cooling fins. You do not have to bend up the electrode.

Ignition module gap
Place a business or index card between the fly wheel and coil, rotate until the magnetic attraction pulls everything tight together. Tighten the coil screws, rotate the flywheel until the attraction eases and remove the card. Ensure nothing contacts when the flywheel is rotated.

If you have more questions or need additional help please reply below and I will get back to you. HTH & Good Luck.
Lou
Thank You for using FixYa.

Immediately check for spark at the plug. If nothing, try a new plug first. If still nothing, remove the stop switch connection at the ignition module and try for spark again. If still nothing, it's new ignition module time. Use a thin business card for the core gap spacer, turn the magnets up under the new module and allow the module to stick to the magnets. Tighten the module screws and rotate the flywheel magnets out from under the module to remove the card. Turn the flywheel one turn to make sure the gap doesn't close up at some point. Hope this helps!