A technique is described for measuring electrical currents in the solar corona. It uses radioastronomical polarization measurements of a spatially-extended radio source viewed through the corona. The observations yield the difference in the Faraday rotation measure between two closely-spaced lines of sight through the corona, a measurement referred to as {\em differential Faraday rotation}. It is shown that the expression for differential Faraday rotation is proportional to the path integral $\oint n \vec{B}\cdot \vec{ds}$ where $n$ is the plasma density and $\vec{B}$ is the coronal magnetic field. The integral is around a closed loop (Amperian Loop) in the corona. If the plasma density is assumed roughly constant, the differential Faraday rotation is proportional to the current within the loop, via Ampere's Law. The validity of the constant density approximation is discussed, and two test cases are presented in which the associated error in the inferred current is small, of order tens of percent or less. The method is illustrated with observations of the radio source 3C228 with the Very Large Array (VLA) in August, 2003. A measurement of a differential Faraday rotation ``event'' on August 16, 2003, yields an estimate of $2.5 \times 10^9$ Amperes in the Amperian Loop. A smaller event on August 18 yields an enclosed current of $2.3 \times 10^8$ Amperes. The implications of these currents for coronal heating are briefly discussed.

Revised version of paper; accepted for Astrophysical Journal, vol 670, November 20, 2007