Distributed Acoustic Sensing (DAS) is a new technology in which seismic energy is detected, at high spatial and temporal resolution, using the propagation of laser pulses in a fibre-optic cable. We show analyses from the first glaciological borehole DAS deployment to measure the englacial and subglacial seismic properties of Store Glacier, a fast-flowing outlet of the Greenland Ice Sheet. We record compressional and shear waves in 1043 m-deep vertical seismic profiles, sampled at 10 m vertical resolution, and detect a transition from isotropic to anisotropic ice at 84% of ice thickness, consistent with the Holocene-Wisconsin transition. We identify subglacial reflections originating from the base of a 20 m-thick layer of consolidated sediment and, from attenuation measurements, interpret temperate ice in the lowermost 100 m of the glacier. Our findings highlight the promising potential of DAS technology to constrain the seismic properties of glaciers and ice sheets.