We have measured the polarization of J/psi and psi(2S) mesons produced in pp collisions at sqrt[s] = 1.8 TeV, using data collected at the Collider Detector at Fermilab during 1992-1995. 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) greater than or approximately equal 12 GeV/c we do not observe significant polarization in the prompt component.