An exact solution for the squeeze-film motion in an upper convected Maxwell fluid is given for both the plane and axisymmetric cases. Inertia and viseoelastic effects are included, and it is shown that the solution depends only on the product of the Weissenberg and Reynolds numbers. Solutions are generated for values of this product up to 500 without numerical problems. The solutions show wave propagation and show a reduced load capacity relative to the Newtonian case.