Cross-section values for Compton scattering on the proton were measured at 25 kinematic settings over the range s=5-11 and -t=2-7 GeV2 with a statistical accuracy of a few percent. The scaling power for the s dependence of the cross section at fixed center-of-mass angle was found to be 8.0+/-0.2, strongly inconsistent with the prediction of perturbative QCD. The observed cross-section values are in fair agreement with the calculations using the handbag mechanism, in which the external photons couple to a single quark.