  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 ($\varepsilon$) and momentum transfer
($Q^2$) simultaneously, as well as to cancel luminosity-related systematic
errors. The cross section ratio increases with decreasing $\varepsilon$ at $Q^2
= 1.45 \text{ GeV}^2$. This measurement is consistent with the size of the form
factor discrepancy at $Q^2\approx 1.75$ GeV$^2$ and with hadronic calculations
including nucleon and $\Delta$ intermediate states, which have been shown to
resolve the discrepancy up to $2-3$ GeV$^2$.
