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Inelastic X-ray scattering (IXS) measurements have been performed on an MgO single crystal in order to evaluate IXS as a methodology for accurate and precise determination of elastic constants and sound velocities. By performing the IXS experiment using a 12-analyzer array, the complete set of single-crystal elastic constants of MgO were determined to a precision better than 0.8% (sound velocities to better than 0.2%). The results are consistent with values in the literature. The precision and accuracy of this work, which is significantly better than other published work to date, demonstrates the potential of IXS in determining elastic properties.