Background: The electromagnetic form factors of the proton measured by unpolarized and polarized electron scattering experiments show a significant disagreement that grows with the squared four-momentum transfer (Q2). Calculations have shown that the two measurements can be largely reconciled by accounting for the contributions of two-photon exchange (TPE). TPE effects are not typically included in the standard set of radiative corrections since theoretical calculations of the TPE effects are highly model dependent, and, until recently, no direct evidence of significant TPE effects has been observed. Purpose: We measured the ratio of positron-proton to electron-proton elastic-scattering cross sections in order to determine the TPE contribution to elastic electron-proton scattering and thereby resolve the proton electric form factor discrepancy. Methods: We produced a mixed simultaneous electron-positron beam in Jefferson Lab's Hall B by passing the 5.6-GeV primary electron beam through a radiator to produce a bremsstrahlung photon beam and then passing the photon beam through a convertor to produce electron-positron pairs. The mixed electron-positron (lepton) beam with useful energies from approximately 0.85 to 3.5 GeV then struck a 30-cm-long liquid hydrogen (LH2) target located within the CEBAF Large Acceptance Spectrometer (CLAS). By detecting both the scattered leptons and the recoiling protons, we identified and reconstructed elastic scattering events and determined the incident lepton energy. A detailed description of the experiment is presented. Results: We present previously unpublished results for the quantity R2γ, the TPE correction to the elastic-scattering cross section, at Q2≈0.85 and 1.45 GeV2 over a large range of virtual photon polarization ?. Conclusions: Our results, along with recently published results from VEPP-3, demonstrate a nonzero contribution from TPE effects and are in excellent agreement with the calculations that include TPE effects and largely reconcile the form-factor discrepancy up to Q2≈2GeV2. These data are consistent with an increase in R2γ with decreasing ? at Q2≈0.85 and 1.45 GeV2. There are indications of a slight increase in R2γ with Q2.