  We present a search for "weak" MgII absorbers [those with W_r(2796) < 0.3 A
in the HIRES/Keck spectra of 26 QSOs. We found 30, of which 23 are newly
discovered. The spectra are 80% complete to W_r(2796) = 0.02 A and have a
cumulative redshift path of ~17.2 for the redshift range 0.4 < z < 1.4. The
number of absorbers per unit redshift, dN/dz, is seen to increase as the
equivalent width threshold is decreased; we obtained dN/dz = 1.74+/-0.10 for
our 0.02 <= W_r(2796) < 0.3 A sample. The equivalent width distribution follows
a 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 absorption
population, which outnumbers Lyman limit systems (LLS) by a factor of
3.8+/-1.1; the majority of weak MgII absorbers must arise in sub-LLS
environments. Tentatively, we predict that ~5% of the Lyman-alpha forest clouds
with 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 as
tracers of sub-LLS environments where gas has been processed by stars. We
compare the number density of W_r(2796) > 0.02 A absorbers with that of both
high and low surface brightness galaxies and find a fiducial absorber size of
35h^-1 to 63h^-1 kpc, depending upon the assumed galaxy population and their
absorption properties. The individual absorbing "clouds" have W_r(2796) <= 0.15
A and their narrow (often unresolved) line widths imply temperatures of ~25,000
K. We measured W_r(1548) from CIV in FOS/HST archival spectra and, based upon
comparisons with FeII, found a range of ionization conditions (low, high, and
multi-phase) in absorbers selected by weak MgII.
