The fine-structure constant is determined with an accuracy of 81 parts per trillion using matter-wave interferometry to measure the rubidium atom recoil velocity. The standard model of particle physics is remarkably successful because it is consistent with (almost) all experimental results. However, it fails to explain dark matter, dark energy and the imbalance between matter and antimatter in the Universe. Because discrepancies between standard-model predictions and experimental observations may provide evidence of new physics, an accurate evaluation of these predictions requires highly precise values of the fundamental physical constants. Among them, the fine-structure constant alpha is of particular importance because it sets the strength of the electromagnetic interaction between light and charged elementary particles, such as the electron and the muon. Here we use matter-wave interferometry to measure the recoil velocity of a rubidium atom that absorbs a photon, and determine the fine-structure constant alpha(-1) = 137.035999206(11) with a relative accuracy of 81 parts per trillion. The accuracy of eleven digits in alpha leads to an electron g factor(1,2)-the most precise prediction of the standard model-that has a greatly reduced uncertainty. Our value of the fine-structure constant differs by more than 5 standard deviations from the best available result from caesium recoil measurements(3). Our result modifies the constraints on possible candidate dark-matter particles proposed to explain the anomalous decays of excited states of Be-8 nuclei(4) and paves the way for testing the discrepancy observed in the magnetic moment anomaly of the muon(5) in the electron sector(6).