The Escherichia coli flagellar motor senses the intracellular concentration of the response regulator CheY-P and responds by varying the bias between its counterclockwise (CCW) and clockwise (CW) rotational states. The response is ultrasensitive with a large Hill coefficient (approximately 10). Recently, the detailed distribution functions of the CW and the CCW dwell times have been measured for different CW biases. Based on a general result on the properties of the dwell-time statistics for all equilibrium models, we show that the observed dwell-time statistics imply that the flagellar motor switch operates out of equilibrium, with energy dissipation. We propose a dissipative allosteric model that generates dwell-time statistics consistent with the experimental results. Our model reveals a general nonequilibrium mechanism for ultrasensitivity wherein the switch operates with a small energy expenditure to create high sensitivity. In contrast to the conventional equilibrium models, this mechanism does not require one to assume that CheY-P binds to the CCW and CW states with different affinities. The estimated energy consumption by the flagellar motor switch suggests that the transmembrane proton motive force, which drives the motor's rotation, may also power its switching. The existence of net transitional fluxes between microscopic states of the switch is predicted, measurement of these fluxes can test the nonequilibrium model directly. Both the results on the general properties of the dwell-time statistics and the mechanism for ultrasensitivity should be useful for understanding a diverse class of physical and biological systems.