  We present an investigation of MgII absorbers characterized as single-cloud
weak systems at z~1. We measured column densities and Doppler parameters for
MgII and FeII in 15 systems found in HIRES/Keck spectra at 6.6 km/s. Using
these quantities and CIV, Lyman alpha and Lyman limit absorption observed with
FOS/HST (resolution ~230 km/s) we applied photoionization models to each system
to constrain metallicities, densities, ionization conditions, and sizes.
  We find that: (1) Single-cloud weak systems are optically thin in neutral
hydrogen and may have their origins in a population of objects distinct from
the optically thick strong MgII absorbers, which are associated with bright
galaxies. (2) Weak systems account for somewhere between 25% to 100% of the z <
1 Lyman alpha forest clouds in the range 15.8<log N(HI)<16.8 cm^-2. (3) At
least seven of 15 systems have two or more ionization phases of gas (multiphase
medium). (4) We identify a subset of weak MgII absorber that we term
``iron-rich''. These clouds are not alpha-group enhanced and are constrained to
have sizes of ~10 pc. At that size, to produce the observed redshift path
density, they would need to outnumber L* galaxies by approximately six orders
of magnitude.
  We discuss these results and the implications that the iron-rich systems
require enrichment from Type Ia supernovae. Further, we address how star
clusters or supernova remnants in dwarf galaxies might give rise to absorbers
with the inferred properties. This would imply far larger numbers of such
objects than are presently known, even locally. We compare the weak systems to
the weak kinematic subsystems in strong MgII absorbers and to Galactic high
velocity clouds. (abridged)
