A four component clay-polymer-salt-water system, consisting of n-butylammonium vermiculite, poly(ethylene oxide), n-butylammonium chloride, and heavy water, was studied by neutron scattering. The volume fraction of clay in the system and the salt concentration were held constant, at r = 0.01 and c = 0.1 M, respectively, and the volume fraction of polymer v was varied between 0 and 0.04. The addition of polymer, up to v 0.04, had no effect on the phase transition temperature between the tactoid and gel phases of the clay system. However, even for v values as low as 0.001, the clay plates in the gel phase were more parallel and more regularly spaced than in the system without added polymer. In the gel phase, the lattice constant d along the swelling axis of the clay colloid decreased as a function of the polymer volume fraction, from 12 nm at v = 0 to 6 nm at v = 0.04. This behavior is qualitatively similar to that observed in the vermiculite system with added poly(vinyl methyl ether). Possible mechanisms for the observed contraction are discussed, including the possibility of bridging flocculation of the clay plates. In one experiment, the d value in the gel phase was observed to increase with temperature prior to collapse into the tactoid phase, which is a new effect that seems to depend on cycling of the system between the gel and tactoid phases.