Magnetic resonance electrical impedance tomography (MREIT) is to visualize the internal current density and conductivity of an electrically conductive object. Injecting current through surface electrodes, we measure one component of the induced internal magnetic flux density using an MRI scanner. In order to reconstruct the conductivity distribution inside the imaging object, most algorithms in MREIT have required multiple magnetic flux density data by injecting at least two independent currents. In this paper, we propose a direct method to reconstruct the internal isotropic conductivity with one component of magnetic flux density data by injecting one current into the imaging object through a single pair of surface electrodes. Firstly, the proposed method reconstructs a projected current density which is a uniquely determined current from the measured one-component magnetic flux density. Using a relation between voltage potential and current, based on Kirchhoff's voltage law, the proposed method is designed to use a combination of two loops around each pixel from which to derive an implicit matrix system for determination of the internal conductivity. Results from numerical simulations demonstrate that the proposed algorithm stably determines the conductivity distribution in an imaging slice. We compare the reconstructed internal conductivity distribution using the proposed method with that using a conventional method with agarose gel phantom experiments.