A finite volume technique is presented for the numerical solution of viscoelastic flows. The flow of a differential upper-convected Maxwell (UCM) model fluid through an abrupt expansion has been chosen as a prototype example due to the existence of previous simulations in the literature. The conservation and constitutive equations are Solved using the finite volume method (FVM) in a non-staggered grid with an upwind scheme for the viscoelastic stresses and a hybrid scheme for the velocities. An enhanced-in-speed pressure-correction algorithm is used and a new method for handling the source term of the momentum equations is introduced. Improved accuracy is achieved by a special discretization of the boundary conditions. Stable solutions are found for high Deborah numbers, further extending the range of previous similar simulations with the FVM. The solutions have been verified with grid refinement and show that at high elasticity levels, the domain length must be long enough to accommodate the slow relaxation of high viscoelastic stresses. The FVM is proven quite capable for numerically handling viscoelastic models with low computational cost and its use is recommended as a viable alternative to the solution of viscoelastic problems using a variety of constitutive models.