In the gas furnace, is it normal for the flame to ever go out when the fan is on?

Yes under the correct circumstances. The flame will go out then the blower will keep blowing till all the residual warm air is out of the furnace. Now if the flame goes out while the thermostat is still not satisfied that could be a problem with the furnace.

Thats whats happening. I takes a while for the thermostat to be satisfied. We may have to kick it up way high. This is an intermittant problem. I'm hoping I just have a bad thermostat. I already changed the batteries.

Intermittent problems are alweays the worst to figure out.

Could it be the stat?

Yes it could be. It could also be other things like the furnace itself. Disconnect the R and W wire in the furnace and jump them together with a piece of wire. Then turn the power switch back on and let the furnace run. It it runs without turning off the burner flame then the problem is probably with the thermostst. If the burner cuts off then the problem is probably with the furnace.

Ok, thanks, I'll try that a get back to you

Intermittent problems are alweays the worst to figure out.Brother, don't even get me started, I'm saving it for my book!:p