We report new observations of ZnII and CrII absorption lines in 10 damped \lya systems (DLAs), mostly at redshift $z_{abs} \simgt 2.5$ . By combining these results with those from our earlier survey (Pettini et al. 1994) and other recent data, we construct a sample of 34 measurements (or upper limits) of the Zn abundance relative to hydrogen [Zn/H]; the sample includes more than one third of the total number of DLAs known. The plot of the abundance of Zn as a function of redshift reinforces the two main findings of our previous study. (1) Damped \lya systems are mostly metal-poor, at all redshifts sampled; the column density weighted mean for the whole data set is [Zn/H] $= -1.13 ± 0.38$ (on a logarithmic scale), or approximately 1/13 of solar. (2) There is a large spread, by up to two orders of magnitude, in the metallicities we measure at essentially the same redshifts. We propose that damped \lya systems are drawn from a varied population of galaxies of different morphological types and at different stages of chemical evolution, supporting the idea of a protracted epoch of galaxy formation. At redshifts $z \simgt 2$ the typical metallicity of the damped \lya systems is in agreement with expectations based on the consumption of HI gas implied by the recent measurements of $Ω_{DLA}$ by Storrie-Lombardi et al. (1996a), and with the metal ejection rates in the universe at these epochs deduced by Madau (1996) from the ultraviolet luminosities of high redshift galaxies revealed by deep imaging surveys. There are indications in our data for an increase in the mean metallicity of the damped \lya systems from $z > 3$ to $≈ 2$, consistent with the rise in the comoving star formation rate indicated by the relative numbers of $U$ and $B$ drop-outs in the Hubble Deep Field. Although such comparisons are still tentative, it appears that these different avenues for exploring the early evolution of galaxies give a broadly consistent picture. ; Comment: 51 pages, LaTeX, 9 Postscript Figures. Accepted for publication in the Astrophysical Journal