There is a significant discrepancy between the values of the proton electric form factor, GpE, extracted using unpolarized and polarized electron scattering. Calculations predict that small two-photon exchange (TPE) contributions can significantly affect the extraction of GpE from the unpolarized electron-proton cross sections. We determined the TPE contribution by measuring the ratio of positron-proton to electron-proton elastic scattering cross sections using a simultaneous, tertiary electron-positron beam incident on a liquid hydrogen target and detecting the scattered particles in the Jefferson Lab CLAS detector. This novel technique allowed us to cover a wide range in virtual photon polarization (ε) and momentum transfer (Q2) simultaneously, as well as to cancel luminosity-related systematic errors. The cross section ratio increases with decreasing ε at Q2=1.45 GeV2. This measurement is consistent with the size of the form factor discrepancy at Q2≈1.75 GeV2 and with hadronic calculations including nucleon and Δ intermediate states, which have been shown to resolve the discrepancy up to 2−3 GeV2.