Using the Atacama Large Millimeter/sub-millimeter Array , we have detected CO ( 3 - 2 ) line and far-infrared continuum emission from a galaxy associated with a high-metallicity ( [ M/H ] = -0.27 ) damped Ly- \alpha absorber ( DLA ) at z _ { DLA } = 2.19289 . The galaxy is located 3.5 ^ { \prime \prime } away from the quasar sightline , corresponding to a large impact parameter of 30 kpc at the DLA redshift . We use archival Very Large Telescope-SINFONI data to detect H \alpha emission from the associated galaxy , and find that the object is dusty , with a dust-corrected star formation rate of 110 ^ { +60 } _ { -30 } M _ { \odot } yr ^ { -1 } . The galaxy ’ s molecular mass is large , M _ { mol } = ( 1.4 \pm 0.2 ) \times 10 ^ { 11 } \times ( \alpha _ { \text { CO } } / 4.3 ) \times ( 0.57 / r _ { 31 } ) M _ { \odot } , supporting the hypothesis that high-metallicity DLAs arise predominantly near massive galaxies . The excellent agreement in redshift between the CO ( 3 - 2 ) line emission and low-ion metal absorption ( \sim 40 { km~ { } s } ^ { -1 } ) disfavors scenarios whereby the gas probed by the DLA shows bulk motion around the galaxy . We use Giant Metrewave Radio Telescope H i 21 cm absorption spectroscopy to find that the H i along the DLA sightline must be warm , with a stringent lower limit on the spin temperature of T _ { s } > 1895 \times ( f / 0.93 ) K. The detection of C i absorption in the DLA , however , also indicates the presence of cold neutral gas . To reconcile these results requires that the cold components in the DLA contribute little to the H i column density , yet contain roughly 50 % of the metals of the absorber , underlining the complex multi-phase nature of the gas surrounding high- z galaxies .