We report on 85−101 GHz light curves of the Galactic Center supermassive black hole, Sagittarius A* (Sgr A*), observed in April 2017 with the Atacama Large Millimeter/submillimeter Array (ALMA). This study of high-cadence full-Stokes data provides new measurements of the fractional linear polarization at a 1−2% level resolved in 4 s time segments, and stringent upper limits on the fractional circular polarization at 0.3%. We compare these findings to ALMA light curves of Sgr A* at 212−230 GHz observed three days later, characterizing a steep depolarization of the source at frequencies below about 150 GHz. We obtain time-dependent rotation measure (RM) values, with the mean RM at 85−101 GHz being a factor of two lower than that at 212−230 GHz. Together with the rapid temporal variability of the RM and its different statistical characteristics in both frequency bands, these results indicate that the Faraday screen in Sgr A* is largely internal, with about half of the Faraday rotation taking place inside the inner 10 gravitational radii, contrary to the common external Faraday screen assumption. We then demonstrate how this observation can be reconciled with theoretical models of radiatively inefficient accretion flows for a reasonable set of physical parameters. Comparisons with numerical general relativistic magnetohydrodynamic simulations suggest that the innermost part of the accretion flow in Sgr A* is much less variable than these models predict; in particular, the observed magnetic field structure appears to be coherent and persistent.