  We have measured the polarizations of $\jpsi$ and $\psiprime$ mesons as
functions of their transverse momentum $\pt$ when they are produced promptly in
the rapidity range $|y|<0.6$ with $\pt \geq 5 \pgev$. The analysis is performed
using a data sample with an integrated luminosity of about $800 \ipb$ collected
by the CDF II detector. For both vector mesons, we find that the polarizations
become increasingly longitudinal as $\pt$ increases from 5 to $30 \pgev$. These
results are compared to the predictions of nonrelativistic quantum
chromodynamics and other contemporary models. The effective polarizations of
$\jpsi$ and $\psiprime$ mesons from $B$-hadron decays are also reported.
