We measured the linear polarization of Sagittarius A* to be 7.2 +/- 0.6 % at 230 GHz using the BIMA array with a resolution of 3.6 * 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* 106 \rdm. These new observations when combined with the JCMT observations at 150, 375 and 400 GHz suggest RM=-4.3 +/- 0.1 * 105 \rdm. 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 \msuny. 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*.