Ice-penetrating radar and ice core drilling have shown that large parts of the north-central Greenland ice sheet are melting from below. It has been argued that basal ice melt is due to the anomalously high geothermal flux that has also influenced the development of the longest ice stream in Greenland. Here we estimate the geothermal flux beneath the Greenland ice sheet and identify a 1,200-km-long and 400-km-wide geothermal anomaly beneath the thick ice cover. We suggest that this anomaly explains the observed melting of the ice sheet's base, which drives the vigorous subglacial hydrology and controls the position of the head of the enigmatic 750-km-long northeastern Greenland ice stream. Our combined analysis of independent seismic, gravity and tectonic data implies that the geothermal anomaly, which crosses Greenland from west to east, was formed by Greenland's passage over the Iceland mantle plume between roughly 80 and 35 million years ago. We conclude that the complexity of the present-day subglacial hydrology and dynamic features of the north-central Greenland ice sheet originated in tectonic events that pre-date the onset of glaciation in Greenland by many tens of millions of years.