Axions and axion-like particles appear in well-motivated extensions of the standard model of particle physics and may be the solution to the long-standing puzzle of the dark matter in our Universe. Several new experiments are foreseen in the next decade searching them in a wide range of the parameter space. In the mass region from few to several tens of microelectronvolt, detector sensitivity will be limited by the standard quantum limit of linear amplifiers and a new class of single microwave-photon detector will be needed. We have developed a single photon counter based on the voltage switching of an underdamped Josephson junction that is coupled to a coplanar waveguide. By measuring the switching voltage, we can register single photons at 14 GHz with the rate less than 1 photon per 3000 s.