Troubleshooting a Circuit Breaker?

We have a dedicated panel for our water well. It has a 15 amp breaker which controls a single light fixture which we use for illumination and, with a heat lamp, for warmth to prevent freezing in bad weather. This is a procedure we have used for 10 years since the panel was installed. I turned on the light at the last freeze and when I returned to turn it off, it was not working. I tested the light bulb which was still good and then I replaced the fixture, but that did not correct the problem. At that point I tested the breaker with my multimeter. It read 124 volts until I turned on the light, then it dropped to 11 volts and the bulb wouldn't glow. I replaced the breaker with a new one and have the same drop in voltage with the very minor load of a single 60W bulb. The well pump and it's double 40A breaker are functioning just fine. What can be the problem?