  We measured the linear polarization of Sagittarius A* to be 7.2 +/- 0.6 % at
230 GHzusing the BIMA array with a resolution of 3.6 x 0.9 arcsec. This
confirms the previously reported detection with the JCMT 14-m antenna. Our high
resolution observations demonstrate that the polarization does not arise from
dust but from a synchrotron source associated with Sgr A*. We see no change in
the polarization position angle and only a small change in the polarization
fraction in four observations distributed over 60 days. We find a position
angle 139 +/- 4 degrees that differs substantially from what was found in
earlier JCMT observations at the same frequency. Polarized dust emission cannot
account for this discrepancy leaving variability and observational error as the
only explanations. The BIMA observations alone place an upper limit on the
magnitude of the rotation measure of 2 x 10^6 rad m^-2. These new observations
when combined with the JCMT observations at 150, 375 and 400 GHz suggest RM
=-4.3 +/- 0.1 x 10^5 rad m^-2. This RM may be caused by an external Faraday
screen. Barring a special geometry or a high number of field reversals, this RM
rules out accretion rates greater than ~ 10^-7 M_sun y^-1. This measurement is
inconsistent with high accretion rates necessary in standard advection
dominated accretion flow and Bondi-Hoyle models for Sgr A*. It argues for low
accretion rates as a major factor in the overall faintness of Sgr A*.
