We consider magnetic resonance electrical impedance tomography, which aims to reconstruct the conductivity distribution using the internal current density furnished by magnetic resonance imaging. We show the uniqueness of the conductivity reconstruction with one measurement imposing the Dirichlet boundary condition. We also propose a fast non-iterative numerical algorithm for the conductivity reconstruction using the internal current vector information. The algorithm is mainly based on efficient numerical construction of equipotential lines. The resulting numerical method is stable in the sense that the error of the computed conductivity is linearly proportional to the input noise level and the introduction of internal current data makes the impedance tomography problem well-posed. We present various numerical examples to show the feasibility of using our method.