The results of a new spectroscopic survey of 66 z greater than or similar t
o4 quasars for Damped Lyman-alpha absorption systems are presented. The sea
rch led to the discovery of 26 new DLA candidates which are analysed in ord
er to compute the comoving mass density of neutral gas in a non-zero Lambda
Universe. The possible sources of uncertainty are discussed and the implic
ations of our results for the theories of galaxy formation and evolution ar
e emphasized. A subsequent paper will present details of the calculations s
ummarised here and a more extensive explanation of the consequences of our
observations for the understanding of the nature of DLAs.