We report the design and experimental demonstration of a gyrotron oscillator using a photonic-band-gap (PBG) structure to eliminate mode competition in a highly overmoded resonator. The PBG cavity supports a TE(041)-like mode at 140 GHz and is designed to have no competing modes over a minimum frequency range delta omega/omega of 30% about the design mode. Experimental operation of a PBG gyrotron at 68 kV and 5 A produced 25 kW of peak power in the design mode. No other modes were observed over the full predicted operating range about the design mode. PBG cavities show great promise for applications in vacuum electron devices in the millimeter- and submillimeter-wave bands.