We have measured the polarization of J/psi and psi (2S) mesons
produced in p anti-p collisions at s**(1/2) = 1.8 TeV, using data collected at
CDF during 1992-95. The polarization of promptly produced J/psi [psi(2S)]
mesons is isolated from those produced in B-hadron decay, and measured over
the kinematic range 4[5.5] < P(t) < 20 GeV/c and |y| < 0.6. For P(t) \geq 12
GeV/c we do not observe significant polarization in the prompt component.