  Quasar foreground damped absorbers, associated with HI-rich galaxies allow to
estimate the neutral gas mass over cosmic time, which is a possible indicator
of gas consumption as star formation proceeds. The DLAs and sub-DLAs are
believed to contain a large fraction of neutral gas mass in the Universe. In
Paper I of the series, we present the results of a search for DLAs and sub-DLAs
in the ESO-UVES Advanced Data Products dataset of 250 quasars. Here we use an
unbiased sub-sample of sub-DLAs from this dataset. We build a subset of 122
quasars ranging from 1.5 <z_em < 5.0, suitable for statistical analysis. The
statistical sample is analyzed in conjunction with other sub-DLA samples from
the literature. This makes up a combined sample of 89 sub-DLAs over a redshift
path of $\Delta z=193$. Redshift evolution of the number density and the line
density are derived for sub-DLAs and compared with the LLSs and DLAs
measurements from the literature. The results indicate that these three classes
of absorbers are evolving in the redshift interval 1 < z < 5. The column
density distribution, f(N,z), down to the sub-DLA limit is determined. The
flattening of f_(N,z) in the sub-DLA regime is present in the observations. The
redshift evolution of f_(N,z) down to sub-DLA regime is also presented,
indicating the presence of more sub-DLAs at high-redshift as compared to
low-redshift. f_(N,z) is further used to determine the neutral gas mass
density, Omega_g, at 1.5 < z < 5.0. The complete sample shows that sub-DLAs
contribute 8-20% to the total Omega_g from 1.5 < z < 5.0. In agreement with
previous studies, no evolution of Omega_g is seen from low-redshift to
high-redshift, suggesting that star formation solely cannot explain this
non-evolution and replenishment of gas and/or recombination of ionized gas is
needed. (Abridged)
