Of several dozen galaxies observed spectroscopically that are candidates for having a redshift (z) in excess of seven, only five have had their redshifts confirmed via Lyman α emission, at z = 7.008, 7.045, 7.109, 7.213 and 7.215 (refs 1-4). The small fraction of confirmed galaxies may indicate that the neutral fraction in the intergalactic medium rises quickly at z > 6.5, given that Lyman α is resonantly scattered by neutral gas. The small samples and limited depth of previous observations, however, makes these conclusions tentative. Here we report a deep near-infrared spectroscopic survey of 43 photometrically-selected galaxies with z > 6.5. We detect a near-infrared emission line from only a single galaxy, confirming that some process is making Lyman α difficult to detect. The detected emission line at a wavelength of 1.0343 micrometres is likely to be Lyman α emission, placing this galaxy at a redshift z = 7.51, an epoch 700 million years after the Big Bang. This galaxy's colours are consistent with significant metal content, implying that galaxies become enriched rapidly. We calculate a surprisingly high star-formation rate of about 330 solar masses per year, which is more than a factor of 100 greater than that seen in the Milky Way. Such a galaxy is unexpected in a survey of our size, suggesting that the early Universe may harbour a larger number of intense sites of star formation than expected.