The effect of elasticity on the spreading of a spinning drop of fluid is investigated in the context of lubrication theory. It is shown that the Oldroyd-B constitutive equation permits a solution in which the free surface of the central part of the drop thins uniformly in space. Perturbation results for small effects of elasticity indicate an increased thinning rate of the free surface compared to Newtonian results for the central part of the spinning drop, and that this enhanced thinning rate persists only over a few characteristic relaxation times. Elastic effects in the capillary region near the moving contact line are also investigated by perturbation theory for small elasticity. Two methods for resolving the contact line singularity are chosen: matching the free surface profile to a precursor film of thickness b, and introducing slip at the spinning plate. For the precursor film model, the free surface correction changes character from a net enhancement of the capillary ridge near the contact line for large b, to a negative correction over most of the profile for small b. With the slip model, the free surface correction gives a net enhancement of the capillary ridge for all values of the slip parameter alpha. The difference between the models for thin precursor films or slight slip is explained by examining the manner in which the stress relaxes near the contact line. The results suggest that viscoelastic contact line dynamics may be more sensitive to the local molecular physics than the Newtonian counterparts.