We describe a procedure for measuring the contribution of relativistic positrons to radio synchrotron radiation. The method relies on the fact that synchrotron radiation from particles of one sign (e.g., electrons) is circularly polarized by a small but measurable amount. If, on the other hand, there are equal numbers of relativistic positrons and electrons, the net circular polarization is zero. The method is illustrated through high-accuracy mapping of the circular polarization of the Crab Nebula at 610 MHz. No significant circular polarization was detected: a very conservative limit is 0.05%, and a more realistic one is 0.03%. We calculate the degree of circular polarization expected if only electrons are present, allowing for the reduction in polarization resulting from nonuniformities in the magnetic field along the line of sight and across the telescope beam. This reduction due to field nonuniformity is estimated from measurements of the degree of linear polarization at optical and high radio frequencies with similar angular resolution to the circular polarization measurements. We find that the observed upper limit on the degree of circular polarization is comparable to or below that expected if only electrons radiate. Various explanations of this result are discussed, including (1) a weaker than assumed magnetic field, (2) a field preferentially nearly perpendicular to the line of sight, (3) a field structure of such a type that nonuniformities reduce the degree of circular polarization by more than they reduce the degree of linear polarization, and (4) the presence of relativistic positrons. Although explanation 1 is implausible, possibilities 2 and 3 cannot be excluded. If future observations establish that the degree of circular polarization at 610 MHz is less than 0.01%, a contribution from positrons would be strongly favored.