We have measured the polarization of J/ψ and ψ(2S) mesons produced in pp¯ collisions at s√=1.8TeV, using data collected at the Collider Detector at Fermilab during 1992–1995. The polarization of promptly produced J/ψ [ψ(2S)] mesons is isolated from those produced in B-hadron decay, and measured over the kinematic range 4 [5.5]<20GeV/c and |y|<0.6. For PT≳12GeV/c we do not observe significant polarization in the prompt component.