I checked for the odor of gas with the water heater gas switch turned off and the gas supply on. No gas odor was detected at the burner tube. With a tank full of hot water, the voltage to the connector that attaches to terminal 1 of the gas valve read zero. When the water heater gas switch was turned on, the voltage went to 13+ volts and the igniter started firing. Therefore, the t-stat does appear to be malfunctioning – just as you suggested, Old Biscuit. Thanks again for your help.
Makes me wonder what would happen if this situation occurred for a long period of time in an unoccupied RV. The water supply would eventually be exhausted (by boiling out of the pressure relief valve) and the burner would continue firing in a dry tank. I would think things could get pretty hot under those circumstances – to the point of perhaps starting a fire. I know this problem will serve as a reminder to me to always turn off the heat source to the hot water heater when it is not being used.