A linear model is used to examine the spin-down of a baroclinic eddy under the sea ice. For anticyclonic eddies the ice stress, besides directly spinning down the azimuthal flow within the mixed layer, generates as Ekman divergence that raises the pycnocline near the eddy axis. For eddies of the size of the baroclinic radius of deformation (based on the unperturbed mixed layer depth H and the density jump across the pycnocline), the doming reaches a quasi-stationary state on the frictional time scale T (defined as H/α, where α is the resistance coefficient at the ice-water interface), which generally is of the order of days. Upward entrainment of the fluid across the pycnocline causes a flow convergence below the mixed layer that continues to spin down the deeper flow and flatten the dome. The erosion of the dome, however, occurs over a much longer time scale of (yN)-1T, where y and N are dimensionless parameters characterizing the entrainment rate and the deep stratification. Using realistic parameter values for the polar eddies, this time scale is of the order of a year or longer. The pycnocline dome observed over the Antarctic warm cells is thus likely to survive into the following freezing season and provide a preconditioning for the deep convection in the Weddell Sea.