Using the 1.19-Bev/c antiproton beam recently discovered at the Berkeley Bevatron of the University of California, we have measured the attenuation cross section in beryllium and copper. These cross sections are compared to attenuation measurements made with the same geometry using positive protons of the same incident energy (497 Mev). The measurements were made at cut-off angles θc of 12.7° for copper, and at 18° for beryllium. For both copper and beryllium the measured attenuation cross section for antiprotons is twice that for positive protons, with a statistical error of +/-15%. In addition, for both elements, more than half the attenuation events resulted in one or more fast charged secondary particles (β>=0.75) - probably indicating that annihilations had taken place. The cross section results are: for copper at θc=12.7°, σ-p=1.58+/-0.22, σp+=0.780+/-0.069 and for beryllium at θc=18°, σ-p=0.365+/-0.059, σp+=0.178+/-0.013, where the units are 10-24 cm2. For copper and beryllium, respectively, the average energies in the absorbers were 430 and 455 Mev.