  We predict non-mesoscopic oscillations in the orbital magnetic moment of a
thin semiconductor ring in the quantum Hall effect regime. These oscillations,
which occur as a function of magnetic field because of a competition between
paramagnetic and diamagnetic currents in the ring, are a direct probe of the
equilibrium current distribution in the nonuniform quantum Hall fluid. The
amplitude of the oscillating moment in a thin ring with major radius $R$ and
minor radius (half width) $a$ is approximately $a R e \omega_{\rm c} /c$, where
$\omega_{\rm c}$ is the cyclotron frequency.
