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, DeltaGC the angular extent of the region, Psi l and Psib; the outer scale on which density fluctuations occur, l0; and the gas temperature, Te. The maximum likelihood estimates of these parameters are DeltaGC = 133-80+200 pc, 0.5o <= Psi l <~ 1o, and (l0/1pc)2/3Te-1/2 = 10-7 +/- 0.8. The parameter Psib was not well constrained and we adopt Psib = 0.50. The close correspondence between DeltaGC and Psi l \dgc suggests that the scattering region encloses the GC. As host media for the scattering, we consider the photoionized surface layers of molecular clouds and the interfaces between molecular clouds and the 107 K ambient gas. We are unable to make an unambiguous determination, but we favor the interface model in which the scattering medium is hot (Te ~ 106 K) and dense (ne ~ 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. We predict that pulsars seen through this region will have a dispersion measure of approximately 2000 pc cm-3, of which approximately 1500 pc cm-3 arises from the GC component itself. We stress the uniqueness of the GC scattering region, probably resulting from the high-pressure environment in the GC.