We demonstrate the narrow switching distribution of an underdamped Josephson junction from the zero to the finite voltage state at millikelvin temperatures. We argue that such junctions can be used as ultrasensitive detectors of the electromagnetic field, operating close to the quantum limit:a given initial (zero voltage) state can be driven by an incoming signal to the finite voltage state. The width of the switching distribution at a nominal temperature of aboutT = 10 mK was 4,5 nA, which corresponds to an effective noise temperature of the device below 60 mK.