  We report a NOrthern Extended Millimeter Array (NOEMA) and Atacama Large
Millimeter/submillimeter Array (ALMA) search for redshifted CO emission from
the galaxies associated with seven high-metallicity ([M/H] $\geq -1.03$) damped
Lyman-$\alpha$ absorbers (DLAs) at $z\approx1.64-2.51$. Our observations
yielded one new detection of CO(3-2) emission from a galaxy at $z=2.4604$ using
NOEMA, associated with the $z=2.4628$ DLA towards QSO B0201+365. Including
previous searches, our search results in detection rates of CO emission of
$\approx56^{+38}_{-24}$ % and $\approx11^{+26}_{-9}$ %, respectively, in the
fields of DLAs with ${\rm [M/H]}>-0.3$ and ${\rm [M/H]}<-0.3$. Further, the
HI-selected galaxies associated with five DLAs with [M/H] $>-0.3$ all have high
molecular gas masses, $\gtrsim5\times10^{10}\ {\rm M}_\odot$. This indicates
that the highest-metallicity DLAs at $z\approx2$ are associated with the most
massive galaxies. The newly-identified $z\approx2.4604$ HI-selected galaxy,
DLA0201+365g, has an impact parameter of $\approx7$ kpc to the QSO sightline,
and an implied molecular gas mass of $(5.04\pm0.78)
\times10^{10}\times(\alpha_{\rm CO}/4.36)\times(r_{31}/0.55)\ {\rm M}_\odot$.
Archival Hubble Space Telescope Wide Field and Planetary Camera 2 imaging
covering the rest-frame near-ultraviolet (NUV) and far-ultraviolet (FUV)
emission from this galaxy yield non-detections of rest-frame NUV and FUV
emission, and a $5\sigma$ upper limit of 2.3 M$_\odot$ yr$^{-1}$ on the
unobscured star formation rate (SFR). The low NUV-based SFR estimate, despite
the very high molecular gas mass, indicates that DLA0201+365g either is a very
dusty galaxy, or has a molecular gas depletion time that is around two orders
of magnitude larger than that of star-forming galaxies at similar redshifts.
