  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.
