  The OLYMPUS collaboration reports on a precision measurement of the
positron-proton to electron-proton elastic cross section ratio, $R_{2\gamma}$,
a direct measure of the contribution of hard two-photon exchange to the elastic
cross section. In the OLYMPUS measurement, 2.01~GeV electron and positron beams
were directed through a hydrogen gas target internal to the DORIS storage ring
at DESY. A toroidal magnetic spectrometer instrumented with drift chambers and
time-of-flight scintillators detected elastically scattered leptons in
coincidence with recoiling protons over a scattering angle range of $\approx
20\degree$ to $80\degree$. The relative luminosity between the two beam species
was monitored using tracking telescopes of interleaved GEM and MWPC detectors
at $12\degree$, as well as symmetric M{\o}ller/Bhabha calorimeters at
$1.29\degree$. A total integrated luminosity of 4.5~fb$^{-1}$ was collected. In
the extraction of $R_{2\gamma}$, radiative effects were taken into account
using a Monte Carlo generator to simulate the convolutions of internal
bremsstrahlung with experiment-specific conditions such as detector acceptance
and reconstruction efficiency. The resulting values of $R_{2\gamma}$, presented
here for a wide range of virtual photon polarization $0.456<\epsilon<0.978$,
are smaller than some hadronic two-photon exchange calculations predict, but
are in reasonable agreement with a subtracted dispersion model and a
phenomenological fit to the form factor data.
