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Quasar foreground damped absorbers, associated with HI-rich galaxies allow toestimate the neutral gas mass over cosmic time, which is a possible indicatorof gas consumption as star formation proceeds. The DLAs and sub-DLAs arebelieved to contain a large fraction of neutral gas mass in the Universe. InPaper I of the series, we present the results of a search for DLAs and sub-DLAsin the ESO-UVES Advanced Data Products dataset of 250 quasars. Here we use anunbiased sub-sample of sub-DLAs from this dataset. We build a subset of 122quasars ranging from 1.5 <z_em < 5.0, suitable for statistical analysis. Thestatistical sample is analyzed in conjunction with other sub-DLA samples fromthe literature. This makes up a combined sample of 89 sub-DLAs over a redshiftpath of $\Delta z=193$. Redshift evolution of the number density and the linedensity are derived for sub-DLAs and compared with the LLSs and DLAsmeasurements from the literature. The results indicate that these three classesof absorbers are evolving in the redshift interval 1 < z < 5. The columndensity distribution, f(N,z), down to the sub-DLA limit is determined. Theflattening of f_(N,z) in the sub-DLA regime is present in the observations. Theredshift 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 tolow-redshift. f_(N,z) is further used to determine the neutral gas massdensity, Omega_g, at 1.5 < z < 5.0. The complete sample shows that sub-DLAscontribute 8-20% to the total Omega_g from 1.5 < z < 5.0. In agreement withprevious studies, no evolution of Omega_g is seen from low-redshift tohigh-redshift, suggesting that star formation solely cannot explain thisnon-evolution and replenishment of gas and/or recombination of ionized gas isneeded. (Abridged)

The ESO UVES Advanced Data Products Quasar Sample - II. Cosmological Evolution of the Neutral Gas Mass Density