A linear theory is developed for the spin-up of a compressible fluid, stratified by a spherical gravity field. Numerical results are obtained for the case of strong stratification (BruntVisl frequency N much greater than the rotation frequency 0). The interior flow is solved in terms of a set of angular eigenfunctions which have been obtained numerically. The principal result is that the spin-up is limited to a layer adjacent to the spherical boundary, the thickness of the layer being of the order of L(0/N), where L is the radius of the boundary. The solution is qualitatively similar to that found by Holton (1965), Walin (1969), and Sakurai (1969a, b) for a stratified fluid in a cylinder. The thickness of the spin-up layer diminishes with latitude , the variation being described roughly by the formula L0| sin |/N. For the case of slow continuous spin-up, the Ekman suction velocity has been calculated, and the results show that || = 24 is the dividing angle between suction (|| > 24) and blowing (|| < 24).