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A closed-form expression for the spatial fading correlation function of a uniform rectangular array (URA) in a three-dimensional (3D) multipath channel is derived. The fading correlation function is expressed in terms of both the azimuth and elevation angle of arrival as well as the antenna spacing and geometry of the URA. Verification is achieved by means of computer simulation where the theoretical and simulation results are shown to be in good agreement. Our results demonstrate that azimuth spread (AS) is the primary determinant of the antenna correlation and the impact of the elevation spread is mainly noticeable at low AS values. The results obtained are vital for capacity analysis in multiple-input multiple-output systems, as well as for sensitivity analysis of the antenna array under study.