Two possible explanations for the low luminosity of the supermassive black hole at the center of our galaxy are (1) an accretion rate of order the canonical Bondi value (roughly 10^{-5} solar masses per year), but a very low radiative efficiency for the accreting gas or (2) an accretion rate much less than the Bondi rate. Both models can explain the broad-band spectrum of the Galactic Center. We show that they can be distinguished using the linear polarization of synchrotron radiation. Accretion at the Bondi rate predicts no linear polarization at any frequency due to Faraday depolarization. Low accretion rate models, on the other hand, have much lower gas densities and magnetic field strengths close to the black hole; polarization may therefore be observable at high frequencies. If confirmed, a recent detection of linear polarization from Sgr A$^*$ above 150 GHz argues for an accretion rate of order 10^{-8} solar masses per year, much less than the Bondi rate. This test can be applied to other low-luminosity galactic nuclei.