  We prove uniqueness for minimizers of the weighted least gradient problem
\[\inf \left\lbrace \int_{\Omega} a|Du|: \ \ u\in BV(\Omega), \ \ u|_{\partial
\Omega}=f \right\rbrace.\] The weight function $a$ is assumed to be continuous
and it is allowed to vanish in certain subsets of $\Omega$. Existence is
assumed a priori. Our approach is motivated by the hybrid inverse problem of
imaging electric conductivity from interior knowledge (obtainable by MRI) of
the magnitude of one current density vector field.
