Scattering of positrons by electrons has been investigated by bombarding a beryllium target with 200-Mev positrons and observing the recoil electrons in a diffusion cloud chamber located behind the target. The cloud chamber was in a magnetic field which permitted the yield of recoil electrons to be measured as a function of their energy W. The experiment covered the range 88<=W<=200 Mev. The positron beam also traversed the cloud chamber and the total number of incident positrons was determined by track-counting. The total electron yield for 88<=W<=200 Mev is (113+/-9)% below that predicted by the first-order Bhabha theory. This difference cannot be interpreted until radiative corrections to the theory have been evaluated. A calculation of these corrections which is valid for the conditions of this experiment is not available. The shape of the electron energy spectrum is in good agreement with the Bhabha theory, and inconsistent with the theory if annihilation terms are omitted.