When galaxy formation started in the history of the Universe remains unclear. Studies of the cosmic microwave background indicate that the Universe, after initial cooling (following the Big Bang), was reheated and reionized by hot stars in newborn galaxies at a redshift in the range 6 < z < 14 (ref. 1). Though several candidate galaxies at redshift z > 7 have been identified photometrically, galaxies with spectroscopically confirmed redshifts have been confined to z < 6.6 (refs 4-8). Here we report a spectroscopic redshift of z = 6.96 (corresponding to just 750Myr after the Big Bang) for a galaxy whose spectrum clearly shows Lyman-α emission at 9,682Å, indicating active star formation at a rate of ~10Msolaryr-1, where Msolar is the mass of the Sun. This demonstrates that galaxy formation was under way when the Universe was only ~6 per cent of its present age. The number density of galaxies at z ~ 7 seems to be only 18-36 per cent of the density at z = 6.6.