We consider the spin-up of a two-layer, stably (density) stratified fluid in a rotating container with an axisymmetric sloping base and cylindrical walls. Details of the spin-up readjustment mechanisms are presented under the assumption of small impulsive changes in the rotation rate of the container. It is shown that the relative positions of the density interface and the discontinuity in wall slope determine the qualitative large-time spin-up response of the fluid. The density interface leads to a spin-up readjustment in each of the fluid layers that is essentially independent. However, when the density interface is below the boundary-slope discontinuity, a sub-region of the upper layer is predicted to readjust in an algebraic rather than exponential manner. A detailed sequence of laboratory experiments have been performed to confirm the predictions of the linear spin-up analysis.