The problem of dispersion from a ground-level, instantaneous point source into an Ekman boundary layer is studied. First and second moments of the material are found for all heights, directions and times, considerably extending the results found earlier by Csanady. At long times after release, the axis of centroids is found to deform into, and freeze in, a fixed shape which is translated along an asymptotically parabolic path. Considerable anisotropy is found in the shape of the cloud, which is elongated at every level in a direction ranging 45° on either side of the geostrophic wind, depending on altitude and elapsed time. At long times, sections at all levels align their directions of elongation 45° veered to the geostrophic, and fully 90° to the ground-level wind shear. In general, after moderate times, dispersion of the cloud will be controlled by wind shear, but eventually control of the dispersion across the short axis of the cloud will be regained by the horizontal diffusion.