We constrain the properties of the mechanism(s) responsible for the bulk of the heating of the corona of the Sun by simulating, for the first time, the appearance of the entire solar corona. Starting from full-sphere magnetic field maps for 2000 December 1 and 8, when the Sun was moderately active, we populate nearly 50,000 coronal field lines with quasi-static loop atmospheres. These atmospheres are based on heating flux densities FH that depend in different ways on the loop half-length L, the field strength B at the chromospheric base, the loop expansion with height, and the heating scale height. The best match to X-ray and EUV observations of the corona over active regions and their environs is found for FH ≈ 4 × 1014B1.0±0.3/L1.0±0.5 (in ergs cm-2 s-1 for B in Mx cm-2 and L in cm), while allowing for substantial loop expansion with increasing height, and for a heating scale height that is at least a sizeable fraction of the loop length. This scaling for coronal heating points to DC reconnection at tangential discontinuities as the most likely coronal heating mechanism, provided that the reconnection progresses proportional to the Alfven velocity. The best-fit coronal filling factor equals unity, suggesting that most of the corona is heated most of the time. We find evidence that loops with half-lengths exceeding ~100,000 km are heated significantly more than suggested by the above scaling, possibly commensurate with the power deposited in the open field of coronal holes.