The recent detection of significant linear polarization at mm and sub-mm wavelengths in the spectrum of Sgr A* (if confirmed) will be a useful probe of the conditions within several Schwarzschild radii (rS) of the event horizon at the Galactic Center. Hydrodynamic simulations of gas flowing in the vicinity of this object suggest that the infalling gas circularizes when it approaches within 5-25 rS of the black hole. We suggest that the sub-mm ``excess'' of emission seen in the spectrum of Sgr A* may be associated with radiation produced within the inner Keplerian region and that the observed polarization characteristics provide direct evidence for this phenomenon.