  We present the preliminary results of a calculation of the fragmentation
contribution to charmonium production at large transverse momentum in
$p\bar{p}$ colliders. The fragmentation of gluons and charm quarks is the
dominant direct production mechanism for sufficiently large $p_{\perp}$. We
find that for both $J/\psi$ and $\psi '$ production fragmentation dominates
over the conventional gluon-gluon fusion mechanism for $p_{\perp}$ greater than
about 6 GeV.} Presented at the Workshop on Physics at Current Accelerator and
the Supercollider, Argonne National Laboratory, June 1993.

High Energy Physics - Phenomenology