Measurement of crater depths in Ganymede’s dark terrain have revealed substantial numbers of unusually shallow craters indicative of viscous relaxation [see companion paper: Singer, K.N., Schenk, P. M., Bland, M.T., McKinnon, W.B., (2017). Relaxed impact craters on Ganymede: Regional variations and high heat flow. Icarus, submitted]. These viscously relaxed craters provide insight into the thermal history of the dark terrain: the rate of relaxation depends on the size of the crater and the thermal structure of the lithosphere. Here we use finite element simulations of crater relaxation to constrain the heat flux within the dark terrain when relaxation occurred. We show that the degree of viscous relaxation observed cannot be achieved through radiogenic heating alone, even if all of the relaxed craters are ancient and experienced the high radiogenic fluxes present early in the satellite’s history. For craters with diameter ≥ 10 km, heat fluxes of 40–50 mW m^-2 can reproduce the observed crater depths, but only if the fluxes are sustained for ∼1 Gyr. These craters can also be explained by shorter-lived “heat pulses” with magnitudes of ∼100 mW m^-2 and timescales of 10–100 Myr. At small crater diameters (4 km) the observed shallow depths are difficult to achieve even when heat fluxes as high as 150 mW m^-2 are sustained for 1 Gyr. The extreme thermal conditions required to viscously relax small craters may indicate that mechanisms other than viscous relaxation, such as topographic degradation, are also in play at small crater diameters. The timing of the relaxation event(s) is poorly constrained due to the sparsity of adequate topographic information, though it likely occurred in Ganymede’s middle history (neither recently, nor shortly after satellite formation). The consistency between the timing and magnitude of the heat fluxes derived here and those inferred from other tectonic features suggests that a single event caused both Ganymede’s tectonic deformation and its crater relaxation. Future observations should permit more robust determination of the relative timing of the heating event that caused viscous relaxation.