The seismological properties of the Earth's inner core have become of particular interest as we understand more about its composition and thermal state(1,2). Observations of anisotropy and velocity heterogeneity in the inner core are beginning to reveal how it has grown and whether it convects(3,4). The attenuation of seismic waves in the inner core is strong, and studies of seismic body waves(5,6) have found that this high attenuation is consistent with either scattering or intrinsic attenuations, The outermost portion of the inner core has been inferred to possess layering and to be less anisotropic than at greater depths(7-10). Here we present observations of seismic waves scattered in the inner core which follow the expected arrival time of the body-wave reflection from the inner-core boundary. The amplitude of these scattered waves can be explained by stiffness variations of 1.2% with a scale length of 2 kilometres across the outermost 300 km of the inner core. These variations might be caused by variations in composition, by pods of partial melt in a mostly solid matrix or by variations in the orientation or strength of seismic anisotropy.