  The formation and evolution of galaxies require large reservoirs of cold,
neutral gas. The damped Lya systems (DLAs), seen in absorption towards 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(HI) = 21.1 +/-0.4 cm^-2) DLA at z ~ 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 ~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 (>~10^6 - 10^9
M_solar) 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.
