Bruce - If I had to guess, I would say it might be the starting capacitor on the blower motor.
Here is a link to a similar situation.
If it is this capacitor the blower motor will be very hot. The reason it stops running is that the motor cannot get to full speed and therefore overheats. A thermal shutoff stops the motor from melting itself.
The capacitor should be about a $10-20 part, but much more if you have a serviceman come out.
Hope this helps,
WallyH