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.
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