  We present a new measurement of the inclusive and differential production
cross sections of $J/\psi$ mesons and $b$-hadrons in proton-antiproton
collisions at $\sqrt{s}=1960$ 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 $J/\psi$ production for all transverse
momenta from 0 to 20 GeV/$c$ in the rapidity range $|y|<0.6$ to be $4.08 \pm
0.02 (stat)^{+0.36}_{-0.33} (syst) \mu {\rm b}$. We separate the fraction of
$J/\psi$ events from the decay of the long-lived $b$-hadrons using the lifetime
distribution in all events with $p_T(J/\psi) > 1.25$ GeV/$c$. We find the total
cross section for $b$-hadrons, including both hadrons and anti-hadrons,
decaying to $J/\psi$ with transverse momenta greater than 1.25 GeV/$c$ in the
rapidity range $|y(J/\psi)|<0.6$, is $ 0.330 \pm 0.005 (stat)
^{+0.036}_{-0.033} (syst) ~\mu{\rm b}$. Using a Monte Carlo simulation of the
decay kinematics of $b$-hadrons to all final states containing a $J/\psi$, we
extract the first measurement of the total single $b$-hadron cross section down
to zero transverse momentum at $\sqrt{s}=1960$ 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 \pm 0.4 (stat)^{+2.5}_{-2.3}
(syst) \mu{\rm b}$.
