The exact nature of weak Mg ii absorbers ( those with W _ { r } ( 2796 ) < 0.3 ~ { } \hbox { { \AA } } ) is a matter of debate , but most are likely related to areas of local star formation or supernovae activity outside of giant galaxies . Using 18 QSO spectra obtained with the Ultra-Violet Echelle Spectrograph ( UVES ) on the Very Large Telescope ( VLT ) , we have conducted a survey for weak Mg ii absorbers at 1.4 < z < 2.4 . We searched a redshift path length of \Delta z = 8.51 , eliminating regions badly contaminated by atmospheric absorption so that the survey is close to 100 % complete to W _ { r } ( 2796 ) = 0.02 ~ { } \hbox { { \AA } } . We found a total of 9 weak absorbers , yielding a number density of absorbers of dN / dz = 1.06 \pm 0.12 for 0.02 \leq W _ { r } ( 2796 ) < 0.3 ~ { } \hbox { { \AA } } . Narayanan et al . ( 19 ) found dN / dz = 1.00 \pm 0.20 at 0 < z < 0.3 and Churchill et al . ( 7 ) found dN / dz = 1.74 \pm 0.10 at 0.4 < z < 1.4 . Therefore , the population of weak Mg ii absorbers appears to peak at z \sim 1 . We explore the expected evolution of the absorber population subject to a changing extragalactic background radiation ( EBR ) from z = 0.9 to z = 1.78 ( the median redshift of our survey ) , and find that the result is higher than the observed value . We point out that the peak epoch for weak Mg ii absorption at z \sim 1 may coincide with the peak epoch of global star formation in the dwarf galaxy environment .