  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)x10^5 rad/m^2; the measurements are consistent with a
constant value. We conservatively assign a 3sigma upper limit of 2x10^5 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 2x10^-7
M_sun/yr if the magnetic field is near equipartition, ordered, and largely
radial, while a lower limit of 2x10^-9 M_sun/yr holds even for a
sub-equipartition, disordered, or toroidal field. The mean intrinsic position
angle is 167+/-7 degrees and we detect variations of 31(+18/-9) degrees. These
variations must originate in the submillimeter photosphere, rather than arising
from rotation measure changes.
