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Recent radioastronomical observations of Faraday rotation in the solar corona can be interpreted as evidence for coronal currents, with values as large as 2.5×10<sup>9</sup> Amperes (Spangler, 2007). These estimates of currents are used to develop a model for Joule heating in the corona. It is assumed that the currents are concentrated in thin current sheets, as suggested by theories of two dimensional magnetohydrodynamic turbulence. The Spitzer result for the resistivity is adopted as a lower limit to the true resistivity. The calculated volumetric heating rate is compared with an independent theoretical estimate by Cranmer et al. (2007). This latter estimate accounts for the dynamic and thermodynamic properties of the corona at a heliocentric distance of several solar radii. Our calculated Joule heating rate is less than the Cranmer et al estimate by at least a factor of 3×10<sup>5</sup>. The currents inferred from the observations of Spangler (2007) are not relevant to coronal heating unless the true resistivity is enormously increased relative to the Spitzer value. However, the same model for turbulent current sheets used to calculate the heating rate also gives an electron drift speed which can be comparable to the electron thermal speed, and larger than the ion acoustic speed. It is therefore possible that the coronal current sheets are unstable to current-driven instabilities which produce high levels of waves, enhance the resistivity and thus the heating rate.