  We present a theory of ghost-free massive gravity where the mass of the
graviton is generated through the Brout-Englert-Higgs (BEH) mechanism and one
of the four scalar fields used is that of mimetic gravity. The mass term is not
of the Fierz-Pauli type and the constraint eliminates the Boulware-Deser ghost
which is absent to all orders. We perform a detailed analysis using the methods
of cosmological perturbation theory and consider quantum fluctuations of the
degrees of freedom of massive graviton and mimetic matter. It is shown that for
three of the degrees of freedom of the graviton of mass $m$ the nonlinear
corrections become comparable to the linear terms already at a length scale of
order $m^{-\frac{1}{2}}$. Thus, at smaller scales they become strongly coupled
and the graviton remains with two transverse degrees of freedom which get
strongly coupled only at Planck scale. The mimetic field behaves as cold
particles of half of the graviton mass and could well explain the source of
dark matter in our universe. In the weakly coupled domain mimetic matter is
completely decoupled from the massive graviton.
