  We develop a new method to constrain the physical conditions in the cool
(~10^4 K) circumgalactic medium (CGM) from measurements of ionic column
densities, by assuming that the cool CGM spans a large range of gas densities
and that small high-density clouds are hierarchically embedded in large
low-density clouds. The new method combines the information available from
different sightlines during the photoionization modeling, thus yielding tighter
constraints on CGM properties compared to traditional methods which model each
sightline individually. Applying this new technique to the COS-Halos survey of
low-redshift ~L* galaxies, we find that we can reproduce all observed ion
columns in all 44 galaxies in the sample, from the low-ions to OVI, with a
single universal density structure for the cool CGM. The gas densities span the
range 50 < \rho/\rho_mean < 5x10^5 (\rho_mean is the cosmic mean), while the
physical size of individual clouds scales as ~\rho^-1, from ~35 kpc of the low
density OVI clouds to ~6 pc of the highest density low-ion clouds. The deduced
cloud sizes are too small for this density structure to be driven by
self-gravity, thus its physical origin is unclear. The implied cool CGM mass
within the virial radius is 1.3x10^10 M_sun (~1% of the halo mass), distributed
rather uniformly over the four decades in density. The mean cool gas density
profile scales as R^-1.0, where R is the distance from the galaxy center. We
construct a 3D model of the cool CGM based on our results, which we argue
provides a benchmark for the CGM structure in hydrodynamic simulations. Our
results can be tested by measuring the coherence scales of different ions.
