A core-annular flow of an Oldroyd-B fluid surrounded by a lubricating Bingham fluid is studied using energy stability methods. We consider base flows for which the lubricating fluid is unyielded close to the interface. For small finite restrictions on the size of shear stress and elastic stress perturbations we are able to demonstrate the exponential decay of a suitable energy functional for sufficiently small Reynolds number and Weissenberg number. These results extend those of [1] where only inelastic fluids were considered. (C) 2011 Elsevier B.V. All rights reserved.