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Millimeter-wave emission from the rippled-field magnetron (cross-field free electron laser (FEL)) is investigated experimentally and theoretically. In this device, electrons move in quasi-circular orbits under the combined action of a radial electric field, a uniform axial magnetic field, and a radial azimuthally periodic wiggler magnetic field. In excess of 300 kW of RF power is observed in two narrow spectral lines whose frequency can be tuned continuously from ~25 to ~50 GHz by variation of the axial magnetic field. The observations are interpreted as a FEL type of instability, associated with a resonance in the particle motion of a layer of electrons embedded in the dense spacecharge cloud. The resonance is shown to occur when 2kwÂ¿0 Â¿ (Â¿>0/Â¿0) Â¿1 -(Â¿p/Â¿0)2, where kw is the wiggler wavenumber, Â¿0 is the azimuthal electron velocity, Â¿0 is the relativistic cyclotron frequency in the axial magnetic field, wp is the relativistic plasma frequency, and Â¿0 = [1 - (Â¿0/c)2]-1/2 of the resonant electron layer.