Using the Submillimeter Array, we have made the first high angular resolution measurements of the linear polarization of Sagittarius A* at submillimeter wavelengths, and the first detection of intra-day variability in its linear polarization. We detected linear polarization at 340 GHz (880 um) at several epochs. At the typical resolution of 1.''4*2.''2, the expected contamination from the surrounding (partially polarized) dust emission is negligible. We found that both the polarization fraction and position angle are variable, with the polarization fraction dropping from 8.5% to 2.3% over three days. This is the first significant measurement of variability in the linear polarization fraction in this source. We also found variability in the polarization and total intensity within single nights, although the relationship between the two is not clear from these data. The simultaneous 332 and 342 GHz position angles are the same, setting a one-sigma rotation measure (RM) upper limit of 7*105 rad/m2. From position angle variations and comparison of ``quiescent'' position angles observed here and at 230 GHz we infer that the RM is a few*105 rad/m2, a factor of a few below our direct detection limit. A generalized model of the RM produced in the accretion flow suggests that the accretion rate at small radii must be low, below 10-6-10-7 Mo/yr depending on the radial density and temperature profiles, but in all cases below the gas capture rate inferred from X-ray observations.