Post-starburst (or E+A) galaxies are characterized by low H$alpha$ emission and strong Balmer absorption, suggesting a recent starburst, but little current star formation. Although many of these galaxies show evidence of recent mergers, the mechanism for ending the starburst is not yet understood. To study the fate of the molecular gas, we search for CO (1-0) and (2-1) emission with the IRAM 30m and SMT 10m telescopes in 32 nearby ($0.01<z<0.12$) post-starburst galaxies drawn from the Sloan Digital Sky Survey. We detect CO in 17 (53%). Using CO as a tracer for molecular hydrogen, and a Galactic conversion factor, we obtain molecular gas masses of $M(H_2)=10^{8.6}$-$10^{9.8} M_odot$ and molecular gas mass to stellar mass fractions of $sim10^{-2}$-$10^{-0.5}$, comparable to those of star-forming galaxies. The large amounts of molecular gas rule out complete gas consumption, expulsion, or starvation as the primary mechanism that ends the starburst in these galaxies. The upper limits on $M(H_2)$ for the 15 undetected galaxies range from $10^{7.7} M_odot$ to $10^{9.7} M_odot$, with the median more consistent with early-type galaxies than with star-forming galaxies. Upper limits on the post-starburst star formation rates (SFRs) are lower by $sim10times$ than for star-forming galaxies with the same $M(H_2)$. We also compare the molecular gas surface densities ($Sigma_{rm H_2}$) to upper limits on the SFR surface densities ($Sigma_{rm SFR}$), finding a significant offset, with lower $Sigma_{rm SFR}$ for a given $Sigma_{rm H_2}$ than is typical for star-forming galaxies. This offset from the Kennicutt-Schmidt relation suggests that post-starbursts have lower star formation efficiency, a low CO-to-H$_2$ conversion factor characteristic of ULIRGs, and/or a bottom-heavy initial mass function, although uncertainties in the rate and distribution of current star formation remain.
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