It has been understood since 1897 that accelerating charges must emit electromagnetic radiation. Although first derived in 1904, cyclotron radiation from a single electron orbiting in a magnetic field has never been observed directly. We demonstrate single-electron detection in a novel radio-frequency spectrometer. The relativistic shift in the cyclotron frequency permits a precise electron energy measurement. Precise beta electron spectroscopy from gaseous radiation sources is a key technique in modern efforts to measure the neutrino mass via the tritium decay end point, and this work demonstrates a fundamentally new approach to precision beta spectroscopy for future neutrino mass experiments.