A counter experiment is described which measures the absolute differential electron-electron scattering cross section in the energy interval 0.6 to 1.2 Mev and the absolute differential positron-electron scattering cross section in the energy interval 0.6 to 1.0 Mev. The ratio of these cross sections is also measured with somewhat increased accuracy. The technique of measurement combines good resolution with large energy transfers between the particles to permit a sensitive test of the relativistic features of the M\o{}ller and Bhabha formulas. The results verify the M\o{}ller formula within the 7 percent experimental error. The Bhabha formula is verified within the 10 percent experimental error. The ratio of the M\o{}ller to the Bhabha formula is verified within about 8 percent experimental error.