The zero‐frequency cyclotron wave is studied on an unneutralized, magnetized relativistic‐electron beam. This stationary mode occurs where the Doppler‐shifted beam‐cyclotron dispersion branch crosses the wavenumber axis. In this experiment zero‐frequency‐cyclotron waves have been generated by the injection of foil (and foilless) diode beams into a drift tube. A linear‐cyclotron wave‐dispersion theory is presented which includes the radial electron‐energy variation &ggr;(r) due to beam space charge. The &ggr;(r) variation causes only one propagating eigenmode to exist for each azimuthal‐mode number. Numerical simulations of the beam and the cyclotron wave are described. Experimental measurements of the mode radial dependence, the wavelength scaling with magnetic field, and the wavelength scaling with beam energy are also presented.