A water heater includes a water tank having an inlet and an outlet, and a flue extending through the tank. A nozzle is positioned near a first end of the flue, arranged so as to emit a fuel stream into the flue, and a flame holder is located within the flue in a position to receive the fuel stream and to hold a flame entirely within the flue. A controller variably controls a flow of fuel to the nozzle according to a temperature of water in the tank.