All evidence so far suggests that the absolute spatial orientation of an experiment never affects its outcome. This is reflected in the standard model of particle physics by requiring all particles and fields to be invariant under Lorentz transformations. The best-known tests of this important cornerstone of physics are Michelson-Morley-type experiments verifying the isotropy of the speed of light(1-3). For matter, Hughes-Drever-type experiments(4-11) test whether the kinetic energy of particles is independent of the direction of their velocity, that is, whether their dispersion relations are isotropic. To provide more guidance for physics beyond the standard model, refined experimental verifications of Lorentz symmetry are desirable. Here we search for violation of Lorentz symmetry for electrons by performing an electronic analogue of a Michelson-Morley experiment. We split an electron wave packet bound inside a calcium ion into two parts with different orientations and recombine them after a time evolution of 95 milliseconds. As the Earth rotates, the absolute spatial orientation of the two parts of the wave packet changes, and anisotropies in the electron dispersion will modify the phase of the interference signal. To remove noise, we prepare a pair of calcium ions in a superposition of two decoherence-free states, thereby rejecting magnetic field fluctuations common to both ions(12). After a 23-hour measurement, we find a limit of h x 11 millihertz (h is Planck s constant) on the energy variations, verifying the isotropy of the electron s dispersion relation at the level of one part in 10(18), a 100-fold improvement on previous work(9). Alternatively, we can interpret our result as testing the rotational invariance of the Coulomb potential. Assuming that Lorentz symmetry holds for electrons and that the photon dispersion relation governs the Coulomb force, we obtain a fivefold-improved limit on anisotropies in the speed of light(2,3). Our result probes Lorentz symmetry violation at levels comparable to the ratio between the electroweak and Planck energy scales(13). Our experiment demonstrates the potential of quantum information techniques in the search for physics beyond the standard model.