The linearly polarized millimeter and sub-millimeter emission in Sagittarius A* is produced within 10 Schwarzschild radii of the supermassive black hole at the Galactic Center. We show that the millimeter emission likely originates from a hot accretion disk, where electrons are heated efficiently by turbulent plasma waves. The observed flux density and polarization requires that the disk have an inclination angle of 45^o and its rotation axis be aligned with the major axis of the intrinsic polarization. The disk also needs to be strongly magnetized with a magnetic field energy density comparable to the thermal energy density of the gas. The high flux density and hard spectrum of the sub-millimeter (<1 mm) emission, on the other hand, suggest that it is emitted from small emission regions and therefore associated with flare events occurring either in coronas of the disk or within the last stable orbit. Simultaneous spectrum and polarization measurements will be able to test the model.