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An intense laser pulse in a plasma can accelerate electrons to GeV energies in centimetres. Transverse betatron motion in the plasma wake results in X-ray photons with an energy that depends on the electron energy, oscillation amplitude and frequency of the betatron motion. Betatron X-rays from laser-accelerator electrons have hitherto been limited to spectra peaking between 1 and 10?keV (ref. ). Here we show that the betatron amplitude is resonantly enhanced when electrons interact with the rear of the laser pulse. At high electron energy, resonance occurs when the laser frequency is a harmonic of the betatron frequency, leading to a significant increase in the photon energy. 108 gamma-ray photons, with spectra peaking between 20 and 150?keV, and a peak brilliance >1023?photons?s?1?mrad?2?mm?2 per 0.1% bandwidth, are measured for 700?MeV beams, with 107 photons emitted between 1 and 7?MeV. Femtosecond duration gamma-rays may find uses in imaging, isotope production, probing dense matter, homeland security and nuclear physics. [ABSTRACT FROM AUTHOR]