The flow of a viscoelastic fluid in a co-rotating two-roll mill is studied using a modified Chilcott-Rallison constitutive model, which includes the Hinch-DeGennes formulation of a conformation-dependent friction coefficient. One well known feature of this model, for homogeneous flows, is that there is a hysteresis loop in the dependence of the steady-state configuration on the strain-rate. However, for inhomogeneous flows, al finite polymer concentrations, there is a strong coupling between the velocity and configuration fields, which is shown by present calculations to mask the hysteresis loop, and the associated non-uniqueness of the steady-state configuration. This is a consequence of the reduction in extensional strength of the velocity field with increasing polymeric contribution to the stress field. For small polymer concentrations the transition from the coiled-state to the stretched-state is abrupt in time and the transitional Deborah number range is relatively narrow. However, with increasing polymer concentration, both the time taken to reach the steady-state and the transitional Deborah number range increase significantly. For relatively large values of c, L and De a birefringent pipe-like structure emerges along the outflow axis of the two-roll mill. This and the overall spatial distribution of the configuration qualitatively agree with birefringence photographs for real polymeric liquids.