  We present an extremely deep CO(1-0) observation of a confirmed $z=1.62$
galaxy cluster. We detect two spectroscopically confirmed cluster members in
CO(1-0) with $S/N>5$. Both galaxies have log(${\cal M_{\star}}$/\msol)$>11$ and
are gas rich, with ${\cal M}_{\rm mol}$/(${\cal M_{\star}}+{\cal M}_{\rm
mol}$)$\sim 0.17-0.45$. One of these galaxies lies on the star formation rate
(SFR)-${\cal M_{\star}}$ sequence while the other lies an order of magnitude
below. We compare the cluster galaxies to other SFR-selected galaxies with CO
measurements and find that they have CO luminosities consistent with
expectations given their infrared luminosities. We also find that they have
comparable gas fractions and star formation efficiencies (SFE) to what is
expected from published field galaxy scaling relations. The galaxies are
compact in their stellar light distribution, at the extreme end for all high
redshift star-forming galaxies. However, their SFE is consistent with other
field galaxies at comparable compactness. This is similar to two other sources
selected in a blind CO survey of the HDF-N. Despite living in a highly quenched
proto-cluster core, the molecular gas properties of these two galaxies, one of
which may be in the processes of quenching, appear entirely consistent with
field scaling relations between the molecular gas content, stellar mass, star
formation rate, and redshift. We speculate that these cluster galaxies cannot
have any further substantive gas accretion if they are to become members of the
dominant passive population in $z<1$ clusters.
