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We have used the Australian Square Kilometre Array Pathfinder (ASKAP) radio telescope to search for intervening 21 cm neutral hydrogen (HI) absorption along the line of sight to 53 bright radio continuum sources. Our observations are sensitive to HI column densities typical of Damped Lyman Alpha absorbers (DLAs) in cool gas with an HI spin temperature below about 300-500 K. The six-dish Boolardy Engineering Test Array (BETA) and twelve-antenna Early Science array (ASKAP-12) covered a frequency range corresponding to redshift $0.4<z<1.0$ and $0.37<z<0.77$ respectively for the HI line. Fifty of the 53 radio sources observed have reliable optical redshifts, giving a total redshift path $Δz$ = 21.37. This was a spectroscopically-untargeted survey, with no prior assumptions about the location of the lines in redshift space. Four intervening HI lines were detected, two of them new. In each case, the estimated HI column density lies above the DLA limit for HI spin temperatures above 50-80 K, and we estimate a DLA number density at redshift $z\sim0.6$ of $n(z)=0.19\substack{+0.15 \\ -0.09}$. This value lies somewhat above the general trend of $n(z)$ with redshift seen in optical DLA studies. Although the current sample is small, it represents an important proof of concept for the much larger 21cm First Large Absorption Survey in HI (FLASH) project to be carried out with the full 36-antenna ASKAP telescope, probing a total redshift path $Δz\sim\,50,000$.