  Recent observations have constrained the orbit and structure of the Large
Magellanic Cloud (LMC), implying a well-constrained pericentric passage about
the Milky Way (MW) ~ 50 Myr ago. In this scenario, the LMC's gaseous disk has
recently experienced maximal ram pressure stripping, suggesting the current
extent of its HI disk directly probes the medium in which it is moving. From
the observed stellar and HI distributions of the system we find evidence of a
truncated gas profile along the windward ``leading edge' of the LMC disk,
despite a far more extended stellar component. We explore the implications of
this ram pressure stripping signature, using both analytic prescriptions and
full 3-dimensional hydrodynamic simulations of the LMC. Our simulations subject
the system to a headwind whose velocity components correspond directly to the
recent orbital history of the LMC. We vary the density of this headwind, using
a variety of sampled parameters for a Beta-profile for a theoretical MW
circumgalactic medium (CGM), comparing the resulting HI morphology directly to
observations of the LMC HI and stellar components. This model can match the
radial extent of the LMC's leading (windward) edge only in scenarios where the
MW CGM density at pericentric passage is n(R = 48.2 +/- 5 kpc) = 1.1
(+.44/-.45) x 1e-4 cm^-3. The implied pericentric density proves insensitive to
both the broader CGM structure and temperature profile, thus providing a
model-independent constraint on the local gas density. This result imposes an
important constraint on the density profile of the MW's CGM, and thus the total
baryon content of the MW. From our work, assuming a Beta-profile valid to ~
Rvir, we infer a total diffuse CGM mass M(300 kpc) = 2.6 +/- 1.4 x 1e10 Msun or
approximately 15% of a 1e12 Msun MW's baryonic mass budget.
