We present a study of J/psi and psi(2S) production in p (p) over bar c
ollisions, at root s = 1.8 TeV with the CDF detector at Fermilab. The
J/psi and psi(2S) mesons are reconstructed using their mu(+)mu(-) deca
y modes. We have measured the inclusive production cross section for b
oth mesons as a function of their transverse momentum in the central r
egion, \eta\ < 0.6. We also measure the fraction of these events origi
nating from b hadrons. We thus extract individual cross-sections for J
/psi and psi(2S) mesons from b-quark decays and prompt production. We
find a large excess (approximately a factor of 50) of direct psi(2S) p
roduction compared with predictions from the color singlet model.