A new type of ultra-high-speed single-photon counter for visible and near-infrared wavebands based on an ultrathin NbN hot-electron photodetector (HEP) has been developed. The detector consists of a very narrow superconducting stripe, biased close to its critical current. An incoming photon absorbed by the stripe produces a resistive hotspot and causes an increase in the film's supercurrent density above the critical value, leading to temporary formation of a resistive barrier across the device and an easily measurable voltage pulse. Our NbN HEP is an ultrafast (estimated response time is 30 ps; registered time, due to apparatus limitations, is 150 ps), frequency unselective device with very large intrinsic gain and negligible dark counts. We have observed sequences of output pulses, interpreted as single-photon events for very weak laser beams with wavelengths ranging from 0.5 /spl mu/m to 2.1 /spl mu/m and the signal-to-noise ratio of about 30 dB.