  (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 samples. 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 <z> = 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.
