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We present a new measurement of the inclusive and differential production cross sections of J/ψ mesons and b hadrons in proton-antiproton collisions at √s=1960 GeV. The data correspond to an integrated luminosity of 39.7 pb-1l collected by the CDF run II detector. We find the integrated cross section for inclusive J/ψ production for all transverse momenta from 0 to 20 GeV/c in the rapidity range |y|-0.33+0.36 (syst) µb. We separate the fraction of J/ψ events from the decay of the long-lived b hadrons using the lifetime distribution in all events with pT(J/ψ)>1.25 GeV/c. We find the total cross section for b hadrons, including both hadrons and antihadrons, decaying to J/ψ with transverse momenta greater than 1.25 GeV/c in the rapidity range |y (J/ψ)|-0.33+0.36 (syst) µb. Using a Monte Carlo simulation of the decay kinematics of b hadrons to all final states containing a J/ψ, we extract the first measurement of the total single b-hadron cross section down to zero transverse momentum at √s =1960 GeV. We find the total single b-hadron cross section integrated over all transverse momenta for b hadrons in the rapidity range |y|-2.3+2.5(syst) µb.