The OLYMPUS experiment was designed to measure the ratio between the positron-proton and electron-proton elastic scattering cross sections, with the goal of determining the contribution of two-photon exchange to the elastic cross section. Two-photon exchange might resolve the discrepancy between measurements of the proton form factor ratio, $mu_p G^p_E/G^p_M$, made using polarization techniques and those made in unpolarized experiments. OLYMPUS operated on the DORIS storage ring at DESY, alternating between 2.01~GeV electron and positron beams incident on an internal hydrogen gas target. The experiment used a toroidal magnetic spectrometer instrumented with drift chambers and time-of-flight detectors to measure rates for elastic scattering over the polar angular range of approximately $25^circ$--$75^circ$. Symmetric M{o}ller/Bhabha calorimeters at $1.29^circ$ and telescopes of GEM and MWPC detectors at $12^circ$ served as luminosity monitors. A total luminosity of approximately 4.5~fb$^{-1}$ was collected over two running periods in 2012. This paper provides details on the accelerator, target, detectors, and operation of the experiment.
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