  We compute leading-power fragmentation corrections to $J/\psi$ production at
the Tevatron and the LHC. We find that, when these corrections are combined
with perturbative corrections through next-to-leading order in the strong
coupling constant $\alpha_s$, we obtain a good fit to high-$p_T$ cross section
data from the CDF and CMS Collaborations. The fitted long-distance matrix
elements lead to predictions of near-zero $J/\psi$ polarization in the helicity
frame at large $p_T$.
