We present the first results from an ongoing survey for damped Lyman-α 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(H i) ≥ 20, out of which 6839 have log N(H i) ≥ 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(H i) distribution at ⟨z⟩ = 2.5. Contrary to what is generally believed, the distribution extends beyond 1022 cm-2 with a moderate slope of index ≈-3.5. This result matches the opacity-corrected distribution observed at z = 0 surprisingly well. The cosmological mass density of neutral gas in DLAs is found to be ωgdla ≈ 10-3, evolving only mildly over the past 12 billion years.