The millimeter/submillimeter wavelength polarization of Sgr A* is known to be variable in both magnitude and position angle on time scales down to a few hours. The unstable polarization has prevented measurements made at different frequencies and different epochs from yielding convincing measurements of Faraday rotation in this source. Here we present observations made with the Submillimeter Array polarimeter at 227 and 343 GHz with sufficient sensitivity to determine the rotation measure at each band without comparing position angles measured at separate epochs. We find the 10-epoch mean rotation measure to be (-5.6+/-0.7)*105 rad m-2; the measurements are consistent with a constant value. We conservatively assign a 3 sigma upper limit of 2*105 rad m-2 to rotation measure changes, which limits accretion rate fluctuations to 25%. This rotation measure detection limits the accretion rate to less than 2*10-7 M\sun yr-1 if the magnetic field is near equipartition, ordered, and largely radial, while a lower limit of 2*10-9 M\sun yr-1 holds even for a sub-equipartition, disordered, or toroidal field. The mean intrinsic position angle is 167 DEG+/-7 DEG and we detect variations of 31+18_-9 degrees. These variations must originate in the submillimeter photosphere, rather than arising from rotation measure changes.