The first satellite interferometric measurements of auroral kilometric radiation (AKR) were performed by cross-correlating the waveforms detected by the ISEE 1 and ISEE 2 spacecraft. Such correlations were measured at 125 and 250 kHz for projected ranging from 20 to 3868 km. High correlations were found for all projected baselines, with little or no tendency to decrease even for the longest baselines. These results must be interpreted differently for incoherent and coherent radiation. For incoherent radiation, the correlation as a function of the baseline is the Fourier transform of the source brightness distribution, and this implies an average source region diameter for all of the bursts analyzed of less than about 10 km. For such small source diameters, the required growth rates are too large to be explained by existing incoherent theories, strongly indicating that the radiation must be coherent. For coherent radiation, an upper limit to the source region diameter can be inferred instead from the angular width of the radiation pattern. The close similarity of the spectra at the longest baselines indicates that the angular width of the radiation pattern must be at least 2.5¿, implying that the diameter of the source must be less than about 20 km. At present, the proposed closed-loop radio lasing model is the only known mechanism for providing sources this small.