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