We report the discovery of variability in the linear polarization from the Galactic Center black hole source, Sagittarius A*. New polarimetry obtained with the Berkeley-Illinois-Maryland Association array at a wavelength of 1.3 mm shows a position angle that differs by 28 +/- 5 degrees from observations 6 months prior and then remains stable for 15 months. This difference may be due to a change in the source emission region on a scale of 10 Schwarzschild radii or due to a change of 3* 105\rdm in the rotation measure. We consider a change in the source physics unlikely, however, since we see no corresponding change in the total intensity or polarized intensity fraction. On the other hand, turbulence in the accretion region at a radius 10 to 1000 Rs could readily account for the magnitude and time scale of the position angle change.