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Despite being the dominant force of nature on large scales, gravity remains relatively elusive to precision laboratory experiments. Atom interferometers are powerful tools for investigating, for example, Earth’s gravity, the gravitational constant, deviations from Newtonian gravity and general relativity. However, using atoms in free fall limits measurement time to a few seconds, and much less when measuring interactions with a small source mass. Recently, interferometers with atoms suspended for 70 s in an optical-lattice mode filtered by an optical cavity have been demonstrated. However, the optical lattice must balance Earth’s gravity by applying forces that are a billionfold stronger than the putative signals, so even tiny imperfections may generate complex systematic effects. Thus, lattice interferometers have yet to be used for precision tests of gravity. Here we optimize the gravitational sensitivity of a lattice interferometer and use a system of signal inversions to suppress and quantify systematic efects. We measure the attraction of a miniature source mass to be amass = 33.3 ± 5.6stat ± 2.7syst nm s−2, consistent with Newtonian gravity, ruling out ‘screened ffth force’ theories3,15,16 over their natural parameter space. The overall accuracy of 6.2 nm s−2 surpasses by more than a factor of four the best similar measurements with atoms in free fall. Improved atom cooling and tilt-noise suppression may further increase sensitivity for investigating forces at sub-millimetre ranges, compact gravimetry, measuring the gravitational Aharonov–Bohm effect and the gravitational constant, and testing whether the gravitational field has quantum properties.