An analysis is made for a conjugate heat transfer problem with thermally developing laminar pipe flow, involving two-dimensional wall and axial fluid conduction. The problem is solved numerically by a finite-difference method for a thick walled, two-regional pipe which has constant outside surface temperatures interfaced by a step change. An exact profile is used to discretize the differential equation in the fluid region, and the method exhibited a simple and fast tool to solve this highly complicated problem. The effects of the three defining parameters, the Peclet number, thickness ratio and wall-to-fluid conductivity ratio, are investigated, and it is observed that the most significant of these parameters is the Peclet number, in its effective range (Pe less than or equal to 20).