We consider the free boundary problem of an axisymmetric, cylindrical liquid filament stretching in an extensional flow through a quiescent fluid of negligible viscosity. Our approach provides a systematic framework in which the known Newtonian solution can be generalized to various viscoelastic constitutive models using a condition for the existence of cylindrical solutions. By assuming a power series expansion for the stress, we obtain an analytic solution that describes the filament motion for a viscoelastic filament. We examine this solution in the weakly and strongly viscoelastic limits, as well as in the transient and long time limits. Comparisons of this exact solution with experimental measurements using a viscoelastic polymer solution show strong quantitative agreement. As $t \rightarrow \infty$, both the solution and the observations scale in the Newtonian limit. This transition from viscoelastic to Newtonian scaling provides insight as to how the molecular dynamics of the polymer couple to the filament's motion. This is joint work with Thomas Witelski (Duke), and Andrew Belmonte and Diane Henderson (Penn State).