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−1 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|<0.6 to be 4.08±0.02(stat)+0.36−0.33(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)|<0.6 is 0.330±0.005(stat)+0.036−0.033(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|<0.6 to be 17.6±0.4(stat)+2.5−2.3(syst)  μb.