Flow modification by dilute solutions of polymers in strong elongational flows is analyzed using a two-state model according to which the polymer contribution to the stress balance in the fluid is negligible as long as the polymers are in their coiled state (or are in the process of stretching) and becomes significant as they approach their steady configuration corresponding to an almost fully stretched state. The flow field is separated accordingly into weakly and strongly modified regions and, using mass and momentum conservation considerations, we obtain the flow profiles in these regions. Our prediction of a dip in the velocity and strain rate profiles agrees with experimental observations.