  The large mass of the ninth pseudoscalar meson, the $\eta^\prime$, is
believed to arise from the combined effects of the axial anomaly and the gauge
field topology present in QCD. We report a realistic, 2+1 flavor, lattice QCD
calculation of the $\eta$ and $\eta^\prime$ masses and mixing which confirms
this picture. The physical eigenstates show small octet-singlet mixing with a
mixing angle of $\theta = -14.1(2.8)^\circ$. Extrapolation to physical light
quark mass gives, with statistical errors only, $m_\eta=573(6)$ MeV and
$m_{\eta^\prime}=947(142)$ MeV, consistent with the experimental values of 548
MeV and 958 MeV.
