We examine the behaviour in shear of a viscoelastic fluid model (Johnson-Segalman fluid plus a Newtonian contribution) which exhibits a constitutive instability with respect to shear banding. Because there is a range of stress values at which shear bands can coexist, it is not clear which value of the stress is attained at a given, nominal shear rate, nor whether a selection of some particular value for the stress throughout the unstable range of shear rates exists or not. In this paper, al least for the specific fluid considered, we show that (i) a phase-separated flow actually occurs, and (ii) a selection mechanism for the stress in the shear-banded regime does exist. To obtain these clear-cut results, we used the Couette concentric cylinder geometry, where a ＇seed＇ for the phase separation is automatically provided by the curvature: the portion oi the material which is near to the inner (moving) cylinder is more strongly sheared than any other portion, and can induce the formation of a high shear rate band. At steady state, the existence of a selection mechanism for the stress implies that the ＇volume fraction＇ of the high shear rate phase (not its shear rate) increases by increasing the velocity of the moving cylinder. These and other features of our computed solutions resemble experimental observations. We conclude the paper by showing how a simple variational reasoning can help in locating the selected stress.