  We perform a global leading-order QCD fit to recent polarized structure
function data in order to extract a consistent set of spin-dependent parton
distributions. Assuming that there is no significant intrinsic polarization of
the quark sea, the data are consistent with a modest amount of the proton's
spin carried by the gluon, although the shape of the gluon distribution is not
well constrained. We show how inelastic $J/\psi$ production in polarized
photon-hadron scattering can, in principle, provide definitive information on
the shape of the gluon distribution. (Talk presented by W.J.Stirling at the
27th International Conference on High Energy Physics, Glasgow, July 1994)
