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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 …