Consideration is given to a laminar pipe flow in which the upstream portion of the wall is externally insulated while the downstream portion of the wall is uniformly heated. An analysis of the problem is performed whose special feature is the accounting of axial conduction in both the tube wall and in the fluid. This conjugate heat transfer problem is governed by two dimensionless groups-a wall conductance parameter and the Peclet number, the latter being assigned values from 5 to 50. From numerical solutions, it was found that axial conduction in the wall can carry substantial amounts of heat upstream into the non directly heated portion of the tube. This results in a preheating of both the wall and the fluid in the upstream region, with the zone of preheating extending back as far as twenty radii. The preheating effect is carried downstream with the fluid, raising temperatures all along the tube. The local Nusselt number exhibits fully developed values in the upstream (non directly heated) region as well as in the downstream [directly heated) region. Of the two effects, wall axial conduction can readily overwhelm fluid axial conduction.