Crevasses and englacial fracture networks route meltwater from a glacier's surface to the subglacial drainage system and thus influence glacial hydraulics. However, rapid fracture growth may also lead to sudden and potentially hazardous structural failure of unstable glaciers and ice dams, rifting of ice shelves, or iceberg calving. Here, we use passive seismic recordings to investigate the englacial fracture network on Glacier de la Plaine Morte, Switzerland. Glacier dynamics and the drainage of an ice-marginal lake give rise to numerous icequakes, the majority of which generate dispersed, high-frequency Rayleigh waves. A wide distribution of events allows us to study azimuthal anisotropy between 10 and 30 Hz in order to extract englacial seismic velocities in regions of preferentially oriented crevasses. Beamforming applied to a 100-m-aperture array reveals azimuthal anisotropy of Rayleigh-wave phase velocities reaching a strength of 8% at high frequencies. In addition, we find that the fast direction of wave propagation coincides with the observed surface strike of the narrow crevasses. Forward modeling and inversion of dispersion curves suggest that the azimuthal anisotropy is induced by a 40-m-thick crevassed layer at the surface of the glacier with 8% anisotropy in shear-wave velocity.