Abstract : We present the first results from an ongoing survey for Damped Lyman-alpha systems (DLAs) in the spectra of z>2 quasars observed in the course of the Baryon Oscillation Spectroscopic Survey (BOSS), which is part of the Sloan Digital Sky Survey (SDSS) III. Our full (non-statistical) sample, based on Data Release 9, comprises 12,081 systems with log N(HI)>=20, out of which 6,839 have log N(HI)>=20.3. This is the largest DLA sample ever compiled, superseding that from SDSS-II by a factor of seven. Using a statistical sub-sample and estimating systematics from realistic mock data, we probe the N(HI) distribution at = 2.5. Contrary to what is generally believed, the distribution extends beyond 10^22 cm^-2 with a moderate slope of index\approx-3.5. This result matches surprisingly well the opacity-corrected distribution observed at z = 0. The cosmological mass density of neutral gas in DLAs is found to be Omega_g_DLA~10^-3, evolving only mildly over the past 12 billion years.