Mass loss from the Greenland ice sheet has increased over the past two decades, currently accounting for 25% of global sea-level rise. This is due to increased surface melt driven by atmospheric warming and the retreat and acceleration of marine-terminating glaciers forced by oceanic heat transport. We use ship-based profiles, bathymetric data and moored time series from 2016 to 2017 of temperature, salinity and water velocity collected in front of the floating tongue of the 79 North Glacier in Northeast Greenland. These observations indicate that a year-round bottom-intensified inflow of warm Atlantic Water through a narrow channel is constrained by a sill. The associated heat transport leads to a mean melt rate of 10.4 ± 3.1 m yr–1 on the bottom of the floating glacier tongue. The interface height between warm Atlantic Water and colder overlying water above the sill controls the ocean heat transport’s temporal variability. Historical hydrographic data show that the interface height has risen over the past two decades, implying an increase in the basal melt rate. Additional temperature profiles at the neighbouring Zachariæ Isstrøm suggest that ocean heat transport here is similarly controlled by a near-glacier sill. We conclude that near-glacier, sill-controlled ocean heat transport plays a crucial role for glacier stability.