The transport of electrically charged quasiparticles (based on electrons or ions) plays a pivotal role in modern technology as well as in determining the essential functions of biological organisms. In contrast, the transport of magnetic charges has barely been explored experimentally, mainly because magnetic charges, in contrast to electric ones, are generally considered at best to be convenient macroscopic parameters, rather than well-defined quasiparticles. However, it was recently proposed that magnetic charges can exist in certain materials in the form of emergent excitations that manifest like point charges, or magnetic monopoles. Here we address the question of whether such magnetic charges and their associated currents-'magnetricity'-can be measured directly in experiment, without recourse to any material-specific theory. By mapping the problem onto Onsager's theory of electrolytes, we show that this is indeed possible, and devise an appropriate method for the measurement of magnetic charges and their dynamics. Using muon spin rotation as a suitable local probe, we apply the method to a real material, the 'spin ice' Dy2Ti2O7 (refs 5-8). Our experimental measurements prove that magnetic charges exist in this material, interact via a Coulomb potential, and have measurable currents. We further characterize deviations from Ohm's law, and determine the elementary unit of magnetic charge to be 5  B Å-1, which is equal to that recently predicted using the microscopic theory of spin ice. Our measurement of magnetic charge and magnetic current establishes an instance of a perfect symmetry between electricity and magnetism.