We have measured the polarization of J/psi and psi(2S) mesons produced in p
(p) over bar collisions at root s = 1.8 TeV. using data collected at the C
ollider Detector at Fermilab during 1992-1995. The polarization of promptly
produced J/psi [psi(2S)] mesons is isolated from those produced in B-hadro
n decay, and measured over the kinematic range 4 [5.5] < P-T < 20 GeV/c and
\y\ < 0.6. For P-T greater than or similar to 12 GeV/c we do not observe s
ignificant polarization in the prompt component.