(Abridged) We present the first results from the Complete Optical and Radio Absorption Line System (CORALS) survey. We have compiled a homogeneous sample of radio-selected QSOs from the Parkes Catalogue and searched for damped Lyman alpha systems (DLAs) towards every target, irrespective of its optical magnitude. This approach circumvents selection effects -- particularly from intervening dust -- which have long been suspected to affect DLA surveys in optically-selected, magnitude-limited QSO

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... mples. The CORALS data set consists of 66 z_em > 2.2 QSOs in which 22 DLAs with absorption redshifts 1.8 < z_abs < z_em have been identified over a total redshift interval Delta z = 55.46. In this first paper of the CORALS series we describe the sample, present intermediate resolution spectroscopy and determine the population statistics of DLAs. We deduce a value of the neutral gas mass density traced by DLAs (expressed as a fraction of the closure density) log Omega_DLA h = -2.59^+0.17_-0.24, and a number density of DLAs per unit redshift n(z) = 0.31^+0.09_-0.08, both at a mean redshift = 2.37. Taking into account the errors, we conclude that dust-induced bias in previous surveys may have led to an underestimate of these quantities by at most a factor of two and we have not uncovered a previously unrecognised population of high column density DLAs in front of faint QSOs.