The final stages of the spreading of nonvolatile polymer droplets on smooth solid surfaces have been studied by the technique of x-ray reflectivity. Clean and silanated silicon wafers which are substrates corresponding, respectively, to high and low positive spreading parameters S, have been investigated. In both cases the equilibrium state has been characterized and compared with recent theoretical predictions. We show that for a high S value, the final state cannot be viewed as a dense homogeneous film, whereas for a low S value, the theoretical "pancake" picture could be appropriate.