We present results from an efficient , non-traditional survey to discover damped Ly \alpha ( DLA ) absorption systems with neutral hydrogen column densities N _ { HI } \geq 2 \times 10 ^ { 20 } atoms cm ^ { -2 } and redshifts z < 1.65 . In the past , identification of DLA systems at z < 1.65 has been difficult due to their rare incidence and the need for UV spectroscopy to detect Ly \alpha absorption at these low redshifts . Our survey relies on the fact that all known DLA systems have corresponding Mg ii absorption . In turn , Mg ii absorption systems have been well-studied and their incidence at redshifts 0.1 < z < 2.2 as a function of the Mg ii rest equivalent width , W _ { 0 } ^ { \lambda 2796 } , is known ( Steidel & Sargent 1992 ) . Therefore , by observing the Ly \alpha line corresponding to identified low-redshift Mg ii systems and determining the fraction of these that are damped , we have been able to infer the statistical properties of the low-redshift DLA population . In an earlier paper ( Rao , Turnshek , & Briggs 1995 ) , we presented initial results from an archival study with data from HST and IUE . Now , with new data from our HST GO program , we have more than doubled the sample of Mg ii systems with available ultraviolet spectroscopic data . In total we have uncovered 12 DLA lines in 87 Mg ii systems with W _ { 0 } ^ { \lambda 2796 } \geq 0.3 Å . Two more DLA systems were discovered serendipitously in our HST spectra . At the present time the total number of confirmed DLA systems at redshifts z < 1.65 is 23 . The significant results of the survey are : ( 1 ) the DLA absorbers are drawn almost exclusively from the population of Mg ii absorbers which have W _ { 0 } ^ { \lambda 2796 } \geq 0.6 Å . Moreover , half of all absorption systems with both Mg ii W _ { 0 } ^ { \lambda 2796 } and Fe ii W _ { 0 } ^ { \lambda 2600 } \geq 0.5 Å are DLA systems . ( 2 ) The incidence of DLA systems per unit redshift , n _ { DLA } , decreases as a function of decreasing redshift . The low redshift data are consistent with the larger incidence of DLA systems seen at high redshift ( Wolfe et al . 1995 ) and the inferred low incidence for DLA at z = 0 derived from 21 cm observations of gas-rich spirals ( Rao , Turnshek , & Briggs 1995 ) . However , the errors in our determination are large enough that it is not clear if the decrease per comoving volume begins to be significant at z \approx 2 , or possibly does not set in until z \approx 0.5 . ( 3 ) On the other hand , the cosmological mass density of neutral gas in low-redshift DLA absorbers , \Omega _ { DLA } , is observed to be comparable to that observed at high redshift . In particular , there is no observed trend which would indicate that \Omega _ { DLA } at low redshift is approaching the value at z = 0 , which is a factor of \approx 4 - 6.5 lower than \Omega _ { DLA } . ( 4 ) The low-redshift DLA absorbers exhibit a larger fraction of very high column density systems in comparison to determinations at both high redshift and at z = 0 . In addition , at no redshift is the column density distribution of DLA absorbers observed to fall off in proportion to \sim N _ { HI } ^ { -3 } with increasing column density , a trend that is theoretically predicted for disk-like systems . We discuss this and other mounting evidence that DLA absorption does not arise solely in luminous disks but in a mixture of galaxy types . Although we have doubled the sample of confirmed low-redshift DLA systems , we are still confronted with the statistics of small numbers . As a result , the errors in the low-redshift determinations of n _ { DLA } and \Omega _ { DLA } are substantial . Therefore , aside from the above evolutionary trends , we also discuss associated limitations caused by small number statistics and the robustness of our results . In addition , we note concerns due to gravitational lensing bias , reliance on the Mg ii statistics , dust obscuration , and the sensitivity of local H i 21 cm emission surveys .