  The ratio of the elastic $e^+ p$ to $e^- p$ scattering cross sections has
been measured precisely, allowing the determination of the two-photon exchange
contribution to these processes. This neglected contribution is believed to be
the cause of the discrepancy between the Rosenbluth and polarization transfer
methods of measuring the proton electromagnetic form factors. The experiment
was performed at the VEPP-3 storage ring at beam energies of 1.6 and 1.0 GeV
and at lepton scattering angles between $15^\circ$ and $105^\circ$. The data
obtained show evidence of a significant two-photon exchange effect. The results
are compared with several theoretical predictions.
