We measured the Newtonian constant of gravity, G, using a gravity gradiometer based on atom interferometry. The gradiometer measures the differential acceleration of two samples of laser-cooled Cs atoms. The change in gravitational field along one dimension is measured when a well-characterized Pb mass is displaced. Here, we report a value of G = 6.693 x 10(-11) cubic meters per kilogram second squared, with a standard error of the mean of +/-0.027 x 10(-11) and a systematic error of +/-0.021 x 10(-11) cubic meters per kilogram second squared. The possibility that unknown systematic errors still exist in traditional measurements makes it important to measure G with independent methods.