For a given re-entrant corner geometry, we describe a. two parameter family of solutions for the local asymptotic behaviour of the flow and stress fields of UCM fluids. The two parameters used are the coefficients of the upstream wall shear rate and pressure gradient. In describing this parametric solution, the relationship between the Cartesian and natural stress basis is explained, reconciling these two equivalent formulations for the problem. The asymptotic solution structure investigated here comprises an outer (core) region together with inner regions (single wall boundary layers) located at the upstream and downstream walls. It is implicitly assumed that there are no regions of recirculation at the upstream wall, i.e. we consider flow in the absence of a lip vortex. The essential feature of the analysis is a full description of the matching between the outer and inner regions in both the Cartesian and natural stress bases, as well as the derivation of numerical estimates of important solution parameters such as the coefficients of the stream function and extra-stresses in the outer (core flow) region together with the downstream wall shear rate. This work is divided into two papers, the first one describing the solution structure in the Cartesian stress basis for the core and upstream boundary layer, and the second paper using the natural stress basis which allows the downstream boundary layer solution to be linked through the core to the upstream boundary layer solution. It is the latter formulation of this problem which allows a complete solution description. (C) 2007 Elsevier B.V. All fights reserved.