The flow of ice is temperature-dependent, but direct measurements of englacial temperature are sparse. The dielectric attenuation of radio waves through ice is also temperature-dependent, and radar sounding of ice sheets is sensitive to this attenuation. Here we estimate depth-averaged radar-attenuation rates within the Greenland Ice Sheet from airborne radar-sounding data and its associated radiostratigraphy. Using existing empirical relationships between temperature, chemistry, and radar attenuation, we then infer the depth-averaged englacial temperature. The dated radiostratigraphy permits a correction for the confounding effect of spatially varying ice chemistry. Where radar transects intersect boreholes, radar-inferred temperature is consistently higher than that measured directly. We attribute this discrepancy to the poorly recognized frequency dependence of the radar-attenuation rate and correct for this effect empirically, resulting in a robust relationship between radar-inferred and borehole-measured depth-averaged temperature. Radar-inferred englacial temperature is often lower than modern surface temperature and that of a steady state ice-sheet model, particularly in southern Greenland. This pattern suggests that past changes in surface boundary conditions (temperature and accumulation rate) affect the ice sheet's present temperature structure over a much larger area than previously recognized. This radar-inferred temperature structure provides a new constraint for thermomechanical models of the Greenland Ice Sheet.