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We report a survey of molecular gas in galaxies in the XMMXCS J2215.9-1738 cluster at $z=1.46$. We have detected emission lines from 17 galaxies within a radius of $R_{200}$ from the cluster center, in Band 3 data of the Atacama Large Millimeter/submillimeter Array (ALMA) with a coverage of 93 -- 95 GHz in frequency and 2.33 arcmin$^2$ in spatial direction. The lines are all identified as CO $J$=2--1 emission lines from cluster members at $z\sim1.46$ by their redshifts and the colors of their optical and near-infrared (NIR) counterparts. The line luminosities reach down to $L'_{\rm CO(2-1)}=4.5\times10^{9}$ K km s$^{-1}$ pc$^2$. The spatial distribution of galaxies with a detection of CO(2--1) suggests that they disappear from the very center of the cluster. The phase-space diagram showing relative velocity versus cluster-centric distance indicates that the gas-rich galaxies have entered the cluster more recently than the gas-poor star-forming galaxies and passive galaxies located in the virialized region of this cluster. The results imply that the galaxies have experienced ram-pressure stripping and/or strangulation during the course of infall towards the cluster center and then the molecular gas in the galaxies at the cluster center is depleted by star formation.