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HI-absorption-selected galaxies associated with damped Ly-alpha absorbers (DLAs) provide a unique opportunity to study the gas in .typical. galaxies at high redshifts comma without a luminosity bias. We have used ALMA and NOEMA to detect CORCHETE_OPENCIICORCHETE_CLOSE-158 micron or CO emission from 13 DLA galaxies at z virgul 2-4 comma and the JVLA to detect CO(2-1) or CO(1-0) emission from three of these galaxies. We propose to continue this successful ALMA+JVLA project comma with (1) a JVLA search for CO(1-0) or CO(2-1) emission from five DLA galaxies at zvirgul2-4 with ALMA-NOEMA detections of CO or CORCHETE_OPENCIICORCHETE_CLOSE-158 micron emission comma and (2) an HST ACS-WFC3 imaging of 3 of the 5 DLA galaxies comma covering the rest-frame near-UV continuum. The JVLA observations will yield molecular gas masses for all five galaxies comma and CO excitation information for four systems comma while the HST observations will measure the dust-unobscured star formation rates the combination of ALMA-NOEMA comma JVLA and HST data will allow us to obtain the most complete picture of high-z DLA galaxies to date comma and to directly test whether such galaxies might have been missed in emission-selected samples due to dust obscuration. We request 36.5 hours of JVLA time and 3 orbits of HST time.