The discovery of superconductivity at 200 kelvin in the hydrogen sulfide system at high pressures demonstrated the potential of hydrogen-rich materials as high-temperature superconductors. Recent theoretical predictions of rare-earth hydrides with hydrogen cages and the subsequent synthesis of LaH₁₀ with a superconducting critical temperature (T_c) of 250 kelvin have placed these materials on the verge of achieving the long-standing goal of room-temperature superconductivity. Electrical and X-ray diffraction measurements have revealed a weakly pressure-dependent T_c for LaH₁₀ between 137 and 218 gigapascals in a structure that has a face-centred cubic arrangement of lanthanum atoms. Here we show that quantum atomic fluctuations stabilize a highly symmetrical Fm3m crystal structure over this pressure range. The structure is consistent with experimental findings and has a very large electron-phonon coupling constant of 3.5. Although ab initio classical calculations predict that this Fm3m structure undergoes distortion at pressures below 230 gigapascals, yielding a complex energy landscape, the inclusion of quantum effects suggests that it is the true ground-state structure. The agreement between the calculated and experimental T_c values further indicates that this phase is responsible for the superconductivity observed at 250 kelvin. The relevance of quantum fluctuations calls into question many of the crystal structure predictions that have been made for hydrides within a classical approach and that currently guide the experimental quest for room-temperature superconductivity. Furthermore, we find that quantum effects are crucial for the stabilization of solids with high electron-phonon coupling constants that could otherwise be destabilized by the large electron-phonon interaction, thus reducing the pressures required for their synthesis.