Buoyancy-driven motion of a viscoelastic drop within an unbounded Newtonian fluid domain is examined in the low Reynolds number limit. Domain perturbation is used to obtain analytical results for the steady shape and migration velocity of the drop when the rheological behavior of the drop phase is modeled using the Oldroyd-B or FENE-CR constitutive equations. The drop is found to deform into an oblate spheroid. The resulting shape deformations tend to increase the drag on the drop, while elastic effects are found to have the opposite effect. These results are in good qualitative agreement with earlier results obtained using a third-order fluid model [Sostarecz and Belmonte, J. Fluid Mech. 497:235-252, 2003]. The stability of the drop shape subject to axisymmetric initial perturbations is examined numerically using a combination of the boundary integral and finite volume methods.