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We present a search for "weak" MgII absorbers [those with W_r(2796) < 0.3 Ain the HIRES/Keck spectra of 26 QSOs. We found 30, of which 23 are newlydiscovered. The spectra are 80% complete to W_r(2796) = 0.02 A and have acumulative redshift path of ~17.2 for the redshift range 0.4 < z < 1.4. Thenumber of absorbers per unit redshift, dN/dz, is seen to increase as theequivalent width threshold is decreased; we obtained dN/dz = 1.74+/-0.10 forour 0.02 <= W_r(2796) < 0.3 A sample. The equivalent width distribution followsa power law with slope -1.0; there is no turnover down to W_r(2796) = 0.02 A at<z> = 0.9. Weak absorbers comprise at least 65% of the total MgII absorptionpopulation, which outnumbers Lyman limit systems (LLS) by a factor of3.8+/-1.1; the majority of weak MgII absorbers must arise in sub-LLSenvironments. Tentatively, we predict that ~5% of the Lyman-alpha forest cloudswith W_r(1215) > 0.1 A will have detectable MgII absorption to W_r,min(2796) =0.02 A and that this is primarily a high-metallicity selection effect (Z/Z_sun]> -1). This implies that MgII absorbing structures figure prominently astracers of sub-LLS environments where gas has been processed by stars. Wecompare the number density of W_r(2796) > 0.02 A absorbers with that of bothhigh and low surface brightness galaxies and find a fiducial absorber size of35h^-1 to 63h^-1 kpc, depending upon the assumed galaxy population and theirabsorption properties. The individual absorbing "clouds" have W_r(2796) <= 0.15A and their narrow (often unresolved) line widths imply temperatures of ~25,000K. We measured W_r(1548) from CIV in FOS/HST archival spectra and, based uponcomparisons with FeII, found a range of ionization conditions (low, high, andmulti-phase) in absorbers selected by weak MgII.Comment: Accepted Version: 43 pages, PostScript figures embedded; accepted to  ApJ; updated version includes analysis of CIV absorptio

The Population of Weak Mg ii Absorbers. I. A Survey of 26 QSO HIRES/Keck Spectra