We consider the integration of stresses for the upper convected Maxwell fluid near the corner singularities in driven cavity flow. The velocity field is prescribed as Newtonian. We find that near the upstream corner, the stresses behave like r(-4/3) except near the stationary wall, where the stress grows like r(-2) and near the moving wall, where the stress is infinite along the entire wall. We assume that the downstream corner is well separated from the upstream corner. In that case, the stresses at the downstream corner have only a weak (logarithmic) singularity throughout most of the flow region. Strong growth of stresses occurs as the downstream wall is approached. While stresses grow like r(-2) at the wall, there is a region near the wall where much stronger growth, proportional to r(-4), occurs. Reasons are given why solutions for the full equations of the UCM model (if such solutions exist at all) should be very different from the ones found here where the velocity field is a priori prescribed. (C) 2003 Elsevier Science B.V. All rights reserved.