  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) measurements,
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 of internal character, 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 what these models predict, in particular, the observed
magnetic field structure appears to be coherent and persistent.
