  (abridged) We develop the mid-infrared extinction (MIREX) mapping technique
of Butler & Tan (2009, Paper I), presenting a new method to correct for the
Galactic foreground emission based on observed saturation in independent cores.
Using Spitzer GLIMPSE 8 micron images, this allows us to accurately probe mass
surface densities, Sigma, up to ~0.5g/cm^2 with 2" resolution. We then
characterize the structure of 42 massive starless and early-stage IRDC cores
and their surrounding clumps, measuring Sigma_cl(r) from the core/clump
centers. We first assess the properties of the core/clump at a scale where the
total enclosed mass as projected on the sky is M_cl=60Msun. We find these
objects have a mean radius of R_cl~0.1pc, mean Sigma_cl=0.3g/cm^2 and, if fit
by a power law density profile rho_cl ~ r^{-k_{rho,cl}}, a mean value of
k_{rho,cl}=1.1. If we assume a core is embedded in each clump and subtract the
surrounding clump envelope to derive the core properties, we find a mean core
density power law index of k_{rho,c} = 1.6. We repeat this analysis as a
function of radius and derive the best-fitting power law plus uniform clump
envelope model for each of the 42 core/clumps. The cores have typical masses of
M_c~100Msun and mean Sigma_c~0.1g/cm^2, and are embedded in clumps with
comparable mass surface densities. We conclude massive starless cores exist and
are well-described by singular polytropic spheres. Their relatively low values
of Sigma and the fact that they are IR dark may imply that their fragmentation
is inhibited by magnetic fields rather than radiative heating. Comparing to
massive star-forming cores, there is tentative evidence for an evolution
towards higher densities and steeper density profiles as star formation
proceeds.
