We propose and analyze a regenerative-amplifier free-electron laser (FEL) to produce fully coherent, hard x-ray pulses. The method makes use of narrow-bandwidth Bragg crystals to form an x-ray feedback loop around a relatively short undulator. Self-amplified spontaneous emission (SASE) from the leading electron bunch in a bunch train is spectrally filtered by the Bragg reflectors and is brought back to the beginning of the undulator to interact repeatedly with subsequent bunches in the bunch train. The FEL interaction with these short bunches regeneratively amplifies the radiation intensity and broadens its spectrum, allowing for effective transmission of the x rays outside the crystal bandwidth. The spectral brightness of these x-ray pulses is about 2 to 3 orders of magnitude higher than that from a single-pass SASE FEL.