The electron‐cyclotron maser instability is studied for the case of an anisotropic electron velocity distribution in the regime where the relativistic corrections to the wave dispersion are significant. Solution of the linear dispersion relation reveals that when the plasma frequency–gyrofrequency ratio &ohgr;pe/&OHgr;e<vte/c, the instability is localized just belowk⊥c/&OHgr;e=1. The growth rate is then strongly peaked for emission at 90° to the magnetic field and is considerably larger than would be the case if the cold‐plasma dispersion theory were valid. These features are confirmed by electromagnetic particle simulations. The simulations also show that saturation results from perpendicular diffusion in velocity space and that the saturation level increases as &ohgr;pe/&OHgr;eis decreased. A quasilinear analysis predicts that the saturation level scales as (&OHgr;e/&ohgr;pe)2&ohgr;maxI, where &ohgr;maxIis the maximum linear growth rate. Applications of the maser instability to the generation of the Earth’s auroral kilometric radiation are discussed.