  The polarization of prompt J/psi at the Fermilab Tevatron is calculated
within the nonrelativistic QCD factorization framework. The contribution from
radiative decays of P-wave charmonium states decreases, but does not eliminate,
the transverse polarization at large transverse momentum. The angular
distribution parameter alpha for leptonic decays of the J/\psi is predicted to
increase from near 0 at p_T = 5 GeV to about 0.5 at p_T = 20 GeV. The
prediction is consistent with measurements by the CDF Collaboration at
intermediate values of p_T, but disagrees by about 3 standard deviations at the
largest values of p_T measured.
