We apply the finite element method to calculate the time-dependent velocity of a sphere falling along the axis of a circular cylinder filled with a viscoelastic fluid. The mesh surrounding the sphere translates along the cylindrical wall. For our calculations, we select an Oldroyd-B fluid with viscometric properties similar to those of the M1 Boger fluid. We then calculate the motion of spheres with variable density as one would in the laboratory. We examine the effect of the Weissenberg number and of the geometry upon the essential features of the time-dependent flow. We find that, for a given sphere and a given fluid, the viscoelastic effects are damped when the radius of the cylinder becomes much larger than the radius of the sphere. The velocity overshoot decreases when the Weissenberg number increases. We also examine the sensitivity of the flow to the magnitude of the retardation time.