Efforts to extract a Greenland ice core with a complete record of the Eemian interglacial (130,000 to 115,000 years ago) have until now been unsuccessful. The response of the Greenland ice sheet to the warmer-than-present climate of the Eemian has thus remained unclear. Here we present the new North Greenland Eemian Ice Drilling (`NEEM') ice core and show only a modest ice-sheet response to the strong warming in the early Eemian. We reconstructed the Eemian record from folded ice using globally homogeneous parameters known from dated Greenland and Antarctic ice-core records. On the basis of water stable isotopes, NEEM surface temperatures after the onset of the Eemian (126,000 years ago) peaked at 8 +/- 4 degrees Celsius above the mean of the past millennium, followed by a gradual cooling that was probably driven by the decreasing summer insolation. Between 128,000 and 122,000 years ago, the thickness of the northwest Greenland ice sheet decreased by 400 +/- 250 metres, reaching surface elevations 122,000 years ago of 130 +/- 300 metres lower than the present. Extensive surface melt occurred at the NEEM site during the Eemian, a phenomenon witnessed when melt layers formed again at NEEM during the exceptional heat of July 2012. With additional warming, surface melt might become more common in the future.