  The scattering diameters of Sgr A* and several nearby OH masers (~ 1" at 1
GHz) indicate that a region of enhanced scattering is along the line of sight
to the Galactic center. We combine radio-wave scattering data and free-free
emission and absorption measurements in a likelihood analysis that constrains
the following parameters of the GC scattering region: The GC-scattering region
separation, d; the angular extent of the region, \psi_l; the outer scale on
which density fluctuations occur, l_0; and the gas temperature, T. The maximum
likelihood estimates of these parameters are d = 133_{-80}^{+200} pc, 0.5
degrees <= \psi_l <~ 1 degrees, and (l_0/1 pc)^{2/3}T^{-1/2} = 10^{-7 +/- 0.8}.
As host media for the scattering, we consider the photoionized surface layers
of molecular clouds and the interfaces between molecular clouds and the 10^7 K
ambient gas. We are unable to make an unambiguous determination, but we favor
an interface model in which the scattering medium is hot (T ~ 10^6 K) and dense
(n_e ~ 10 cm^{-3}). The GC scattering region produces a 1 GHz scattering
diameter for an extragalactic source of 90", if the region is a single screen,
or 180", if the region wraps around the GC, as appears probable. We modify the
Taylor-Cordes model for the Galactic distribution of free electrons in order to
include an explicit GC component. Pulsars seen through this region will have a
dispersion measure of approximately 2000 pc cm^{-3}, of which 75% arises from
the GC component. We stress the uniqueness of the GC scattering region,
probably resulting from the high-pressure environment in the GC.
