We report on the discovery of a dual supermassive black hole system in the radio galaxy J1543−0757, with a projected separation between the two black holes of ∼46 mas. The result is based on recent multifrequency observations using the Very Long Baseline Array and European VLBI Network, which reveal two compact, variable, flat-spectrum, active nuclei within the elliptical host galaxy of J1543−0757. Multiepoch observations from the Very Long Baseline Array also provide constraints on the spectral index and proper motions of all components. The flat spectra of both N and S at both frequencies strongly support the hypothesis that these radio components are associated with two separate, accreting supermassive black holes. The two nuclei appear stationary, while the jets emanating from the weaker of the two nuclei appear to move out and terminate in bright hotspots. The discovery of this system has implications for the number of close dual black holes that might be sources of gravitational radiation.