  We consider the problem of recovering an isotropic conductivity outside some
perfectly conducting or insulating inclusions from the interior measurement of
the magnitude of one current density field $|J|$. We prove that the
conductivity outside the inclusions, and the shape and position of the
perfectly conducting and insulating inclusions are uniquely determined (except
in an exceptional case) by the magnitude of the current generated by imposing a
given boundary voltage. We have found an extension of the notion of
admissibility to the case of possible presence of perfectly conducting and
insulating inclusions. This also makes it possible to extend the results on
uniqueness of the minimizers of the least gradient problem
$F(u)=\int_{\Omega}a|\nabla u|$ with $u|_{\partial \Omega}=f$ to cases where
$u$ has flat regions (is constant on open sets).
