We study the flow of an upper convected Maxwell fluid past a cylinder in the limit of high Weissenberg number. We make the simplifying assumption that the velocity field is given and similar to the Newtonian case (as discussed below, this assumption is likely to be wrong) and consider the integration of the stress equations. The asymptotic behavior at high Weissenberg number leads to boundary layers along the edge of the cylinder as well as stress concentration in the wake. The stresses in the boundary layer are of order W while the width of the boundary layer is of order W-1. The stresses in the wake are of order W-3, and the width of the wake is of order W-2. Just outside the boundary layer near the cylinder and wake, there is a region where the stresses are much larger (order W-3 near the cylinder and W-5 near the wake). The origin of these stresses is in the stretching flow near the upstream stagnation point. (C) 2000 Elsevier Science B.V. All rights reserved.