The formation and evolution of galaxies require large reservoirs of cold , neutral gas . The damped Lyman- \alpha systems ( DLAs ) , seen in absorption toward distant quasars and gamma-ray bursts , are predicted to be the dominant reservoirs for this gas . Detailed properties of DLAs have been studied extensively for decades with great success . However , their size , fundamental in understanding their nature , has remained elusive , as quasar and gamma-ray-burst sightlines only probe comparatively tiny areas of the foreground DLAs . Here , we introduce a new approach to measure the full extent of DLAs in the sightlines toward extended background sources . We present the discovery of a high-column-density ( log N ( H i ) = 21.1 \pm 0.4 cm ^ { -2 } ) DLA at z \sim 2.4 covering 90–100 % of the luminous extent of a line-of-sight background galaxy . Estimates of the size of the background galaxy range from a minimum of a few kpc ^ { 2 } , to \sim 100 kpc ^ { 2 } , and demonstrate that high-column density neutral gas can span continuous areas 10 ^ { 8 } –10 ^ { 10 } times larger than previously explored in quasar or gamma-ray burst sightlines . The DLA presented here is the first from a sample of DLAs in our pilot survey that searches Lyman break and Lyman continuum galaxies at high redshift . The low luminosities , large sizes , and mass contents ( \gtrsim 10 ^ { 6 } –10 ^ { 9 } M _ { \odot } ) implied by this DLA and the early data suggest that DLAs contain the necessary fuel for galaxies , with many systems consistent with relatively massive , low-luminosity primeval galaxies .