We measure phonon dispersion and linewidth in a single crystal of MgB_2 along the Gamma-A, Gamma-M and A-L directions using inelastic X-Ray scattering. We use Density Functional Theory to compute the effect of both electron-phonon coupling and anharmonicity on the linewidth, obtaining excellent agreement with experiment. Anomalous broadening of the E_2g phonon mode is found all along Gamma-A. The dominant contribution to the linewidth is always the electron-phonon coupling.