We consider a reconstruction formula for the internal conductivity and uniqueness of conductivity in magnetic resonance electrical impedance tomography (MREIT) which aims to reconstruct the conductivity distribution using internal current distribution. We provide a counter-example of uniqueness for a single measurement of current density with Neumann boundary data and show that at least two measurements are required unless Dirichlet boundary data are given. We present a reconstruction formula and a non-iterative reconstruction method using two internal current densities, which gives a unique conductivity distribution up to a constant factor even without any boundary measurement. The curl-J method is based on the fact that the distortion of the current density vector is induced by the gradient of conductivity orthogonal to the current flow and the fact that no MREIT method can detect the conductivity gradient parallel to the current flow direction directly. We demonstrate the feasibility of our method with several realistic numerical examples.