Using galaxies as background light sources to map the Lya absorption lines is a novel approach to study Damped Lya Absorbers (DLAs). We report the discovery of an intervening z = 3.335 +- 0.007 DLA along a galaxy sight-line identified among 80 Lyman Break Galaxy (LBG) spectra obtained with our Very Large Telescope/Visible Multi-Object Spectrograph survey in the SSA22 field. The measured DLA neutral hydrogen (HI) column density is log (NHI/cm^{-2}) = 21.68 +- 0.17. The DLA covering fraction over the extended background LBG is > 70% (2 sigma), yielding a conservative constraint on the DLA area of > 1 kpc^2. Our search for a counterpart galaxy hosting this DLA concludes that there is no counterpart galaxy with star formation rate (SFR) larger than a few Msun yr^{-1}, ruling out an unobscured violent star formation in the DLA gas cloud. We also rule out the possibility that the host galaxy of the DLA is a passive galaxy with Mstar > 5 x 10^{10} Msun or a heavily dust-obscured galaxy with E(B-V) > 2. The DLA may coincide with a large-scale overdensity of the spectroscopic LBGs. The occurrence rate of the DLA is compatible with that of DLAs found in QSO sight-lines.
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