The unsteady extrusion of a viscoelastic film from an annular and axisymmetric die is examined. External. elastic, viscous and inertia forces deform the film, which is simultaneously cooled via forced convection to the ambient air. This moving boundary problem is solved by mapping the liquid/air interfaces onto fixed ones and by employing a regular perturbation expansion for all the dependent variables. The ratio of the film thickness to its inner radius at the exit of the die is used as the small parameter in the perturbation expansion. The fluid mechanical aspects of the process depend on the Stokes, Deborah, Reynolds, and Capillary numbers. The heat transfer in the film and to the environment gives rise to four additional dimensionless groups: the Peclet, Biot and Brinkman numbers and the activation energy, which determines the temperature dependence of fluid viscosity and elasticity. A variable heat transfer coefficient is also considered. For typical fluid properties and process conditions, the Peclet number is very large. In this case it is the ratio of the Biot to the Peclet number, the Stanton number, which arises in the energy conservation equation. It is shown that film cooling becomes important when the Stanton number and/or the activation energy are in the high-end of their typical values. In such cases, the cooling of the parison leads to a more uniform flow and shape for the film. The influence on the process of a variable heat transfer coefficient and the Brinkman number is small.