There is a significant discrepancy between the values of the proton electric form factor, G(E)(p), extracted using unpolarized and polarized electron scattering. Calculations predict that small two-photon exchange (TPE) contributions can significantly affect the extraction of G(E)(p). 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 (epsilon) and momentum transfer (Q(2)) simultaneously, as well as to cancel luminosity-related systematic errors. The cross section ratio increases with decreasing epsilon at Q(2) = 1.45 GeV2. This measurement is consistent with the size of the form factor discrepancy at Q(2) approximate to 1.75 GeV2 and with hadronic calculations including nucleon and Delta intermediate states, which have been shown to resolve the discrepancy up to 2-3 GeV2.