We have built an atom interferometer that can measure g, the local acceleration due to gravity, with a resolution of Δg/g = 2 × 10−8 after a single 1.3 s measurement cycle, 3 × 10−9 after 1 min and 1 × 10−10 after two days of integration time. The difference between our value for g and one obtained by a falling corner-cube optical interferometer is (7 ± 7) × 10−9 g. The atom interferometer uses velocity-selective stimulated Raman transitions and laser-cooled caesium atoms in an atomic fountain. We extend previous methods of analysing the interferometer to include the effects of a gravitational gradient. We also present detailed experimental and theoretical studies of potential systematic errors and noise sources.