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We present a method to actively outcouple radiation from an x-ray regenerative amplifier. The method uses an undulator within a cavity formed by the diamond mirror and is driven by a high repetition rate linac to build up a high power, narrow bandwidth seed within the cavity. Shaping the electron beam via laser heater modulation on a subsequent pass increases the bandwidth of the x-ray free-electron laser pulse beyond the bandwidth of the diamond mirrors, transmitting the radiation from the cavity. A fraction of the power from the shaped pulse is returned to the cavity, reducing the subsequent number of passes needed to rebuild the seed and increasing the duty factor. The cavity seeding may be used to produce short pulses that are significantly more stable than those produced via self-amplified spontaneous emission (SASE) or enhanced SASE.