We present the first constraint on Faraday rotation measure (RM) at submillimeter wavelengths for the nucleus of M 87. By fitting the polarization position angles ($chi$) observed with the SMA at four independent frequencies around $sim$230 GHz and interpreting the change in $chi$ as a result of emph{external} Faraday rotation associated with accretion flow, we determine the rotation measure of the M 87 core to be between $-$7.5$times$10$^{5}$ and 3.4$times$10$^{5}$ rad/m$^{2}$. Assuming a density profile of the accretion flow that follows a power-law distribution and a magnetic field that is ordered, radial, and has equipartition strength, the limit on the rotation measure constrains the mass accretion rate $dot{M}$ to be below 9.2$times$10$^{-4}$ M$_{odot}$~yr$^{-1}$ at a distance of 21 Schwarzchild radii from the central black hole. This value is at least two orders of magnitude smaller than the Bondi accretion rate, suggesting significant suppression of the accretion rate in the inner region of the accretion flow. Consequently, our result disfavors the classical emph{advection dominated accretion flow} (ADAF) and prefers the emph{adiabatic inflow-outflow solution} (ADIOS) or emph{convection-dominated accretion flow} (CDAF) for the hot accretion flow in M 87.
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