When it quits, you might quickly try and take a volt reading between ground clamp and rod holder. I don't know if there is a thermal breaker in there or not. If there is, it might be opening at too low a temperature. I'm thinking this is most likely.
I appreciate all of the responses. I'm going to try some fresh rod first. The volt meter is a good idea. Maybe I can start at the ends then keep moving down till I find it. It's sitting in the corner. My current problem is too many darn projects.