The asymptotic behaviour of laminar forced convection in a circular tube, for a Newtonian fluid at constant properties, is analysed by taking into account the viscous dissipation effects. The axial heat conduction in the fluid is neglected. A sufficient condition for the existence of a fully developed region is determined. This condition includes, for instance, any asymptotically vanishing axial distribution of the wall heat flux, uniform wall temperature, convection with an external fluid. The asymptotic temperature field and the asymptotic value of the Nusselt number are determined analytically, for every boundary condition which allows a fully developed region. In particular, it is proved that, whenever the wall heat flux tends to zero, the asymptotic Nusselt number is zero.