This paper reports the first convergent numerical algorithm for the steady inertialess flow of an Upper-Convected Maxwell (UCM) fluid through a four-to-one abrupt axisymmetric contraction. The Finite Volume Method (FVM) is adopted along with a stream function-vorticity approach in the Elastic Viscous Split Stress (EVSS) form with a first-order upwind approximation applied to the convective terms of the stress constitutive equation. A staggered volume discretization of the flow variables eliminates the stress singularity from the computational domain without loosing any flow physics. The volume integrals of the governing equations over the flow domain result in a system of nonlinear algebraic equations that are solved iteratively by a semi-implicit line-to-line method with a pseudo-transient term added to the stress constitutive equation. Computations of an UCM fluid using this method are carried out to a much higher value of the Deborah number (De) than previous numerical simulations using the Finite Element Method (FEM). The solutions are found to be smooth, stable, and convergent with the finger stress tensor remaining positive-definite throughout the flow domain. Calculations are not performed above De = 6.25 because of the decreasing pseudo-time step constraint at higher elasticity. The finite volume algorithm approximates better solutions upon mesh refinement, demonstrates the smoothness and the mathematical well-posedness of this problem, and predicts a Newtonian-like flow structure near the singularity for an UCM fluid. Using this new method, the inertialess flow of an UCM fluid through 4:1 abrupt axisymmetric contraction, for the first time, produces larger corner vortices at higher values of De.