The singly differential collision cross sections of the double-photon Compton process have been measured for 662-keV incident gamma photons by using a single-gamma-ray detector. This technique avoids the use of the complicated slow-fast coincidence setup used until now for observing this higher-order process. The measured values of the singly differential collision cross section are of the same magnitude but deviate from the corresponding values calculated from the theory and are nearly (fine-structure constant, [congruent with] 1/137) times the Klein-Nishina cross-section value for different scattering angles.