  We present a calculation of the charm and gluon fragmentation contributions
to inclusive $J/\psi$ and $\psi'$ production at large transverse momentum at
the Tevatron. For $\psi$ production, we include both fragmentation directly
into $\psi$ and fragmentation into $\chi_c$ followed by the radiative decay
$\chi_c \to \psi + \gamma$. We find that fragmentation overwhelms the
leading-order mechanisms for prompt $\psi$ production at large $p_T$, and that
the dominant contributions come from fragmentation into $\chi_c$. Our results
are consistent with recent data on $\psi$ production from the CDF and D0
experiments. In the case of prompt $\psi'$ production, the dominant mechanism
at large $p_T$ is charm fragmentation into $\psi'$. We find serious
disagreement between our theoretical predictions and recent $\psi'$ data from
the Tevatron.
