Most models of cosmic chemical evolution predict that the mass-weighted mean interstellar metallicity of galaxies should rise with time from a low value \sim 1 / 30 solar at z \sim 3 to a nearly solar value at z = 0 . In the absence of any selection effects , the damped Lyman-alpha absorbers ( DLAs ) in quasar spectra are expected to show such a rise in global metallicity . However , it has been difficult to determine whether or not DLAs show this effect , primarily because of the very small number of DLA metallicity measurements at low redshifts . In an attempt to put tighter constraints on the low-redshift end of the DLA metallicity-redshift relation , we have observed Zn II and Cr II lines in four DLAs at 0.09 < z < 0.52 , using the Space Telescope Imaging Spectrograph ( STIS ) onboard the Hubble Space Telescope ( HST ) . These observations have provided the first constraints on Zn abundances in DLAs with z < 0.4 . In all the three DLAs for which our observations offer meaningful constraints on the metallicity , the data suggest that the metallicities are much lower than the solar value . These results are consistent with recent imaging studies indicating that these DLAs may be associated with dwarf or low surface brightness galaxies . We combine our results with higher redshift data from the literature to estimate the global mean metallicity-redshift relation for DLAs . We find that the global mean metallicity shows at most a slow increase with decreasing redshift . For the redshift range 0.09 < z < 3.90 , the slope of the exponential fit to the binned N ( { H I } ) -weighted mean Zn metallicity vs. redshift relation is -0.18 \pm 0.06 counting Zn limits as detections , -0.22 \pm 0.08 counting Zn limits as zeros , and -0.23 \pm 0.06 using constraints on metallicity from other elements in cases of Zn limits . The corresponding estimates of the z = 0 intercept of the metallicity-redshift relation are -0.74 \pm 0.15 , -0.75 \pm 0.18 , and -0.71 \pm 0.13 , respectively . Roughly similar results are obained if survival analysis or an unbinned N ( { H I } ) -weighted nonlinear \chi ^ { 2 } approach is used . Thus , the N ( { H I } ) -weighted mean metallicity of DLAs does not appear to rise up to solar or near-solar values at z = 0 . This weak evolution could be explained by the fact that our absorption-selected sample seems to be dominated by dwarf or low surface brightness galaxies . This suggests that current DLA samples , especially those at low redshifts , could be biased against more enriched galaxies because the latter may cause higher dust obscuration of the background quasars .