We use ground-based and satellite observations to detect large diurnal and longer-period variations in the flow of the Greenland Ice Sheet (GrIS) during late summer that are strongly coupled with changes in its surface hydrology. The diurnal signals are associated with periodic changes in surface melting, and the longer-period signals are associated with the episodic drainage of supra-glacial lakes. Ice velocity doubles around 2 hours after peak daily melting and returns approximately to wintertime levels around 12 hours afterwards, demonstrating an intimate link between the surface and basal hydrology. During late summer, the ice sheet accelerates by 35% per positive degree-day of melting. The observed link between surface melting and enhanced flow is typical of Alpine glaciers, which may provide an appropriate analogue for the evolution of the GrIS in a warming climate.