Studying the flow of baryons into and out of galaxies is an important part of understanding the evolution of galaxies over time. We present a detailed case study of the environment around an intervening Ly α absorption line system at z_abs = 0.633, seen towards the quasar J0423-0130 (z_QSO = 0.915). We detect with ALMA the ¹²CO(2-1), ¹²CO(3-2), and 1.2 mm continuum emission from a galaxy at the redshift of the Ly α absorber at a projected distance of 135 kpc. From the ALMA detections, we infer interstellar medium conditions similar to those in low-redshift luminous infrared galaxies. Director's Discretionary Time (DDT) Multi-Unit Spectroscopic Explorer (MUSE) integral field unit observations reveal the optical counterpart of the ¹²CO emission line source and three additional emission line galaxies at the absorber redshift, which together form a galaxy group. The ¹²CO emission line detections originate from the most massive galaxy in this group. While we cannot exclude that we miss a fainter host, we reach a dust-uncorrected star formation rate (SFR) limit of >0.3 M_☉yr^-1 within 100 kpc from the sightline to the background quasar. We measure the dust-corrected SFR (ranging from 3 to 50 M_☉ yr^-1), the morpho-kinematics and the metallicities of the four group galaxies to understand the relation between the group and the neutral gas probed in absorption. We find that the Ly α absorber traces either an outflow from the most massive galaxy or intragroup gas. This case study illustrates the power of combining ALMA and MUSE to obtain a census of the cool baryons in a bounded structure at intermediate redshift....