A solution is presented for heat transfer from a circular tube to a fluid, with variable properties, in fully developed laminar flow within the tube. The major assumption employed is that the radial velocity component may be neglected. The method is illustrated for supercritical carbon dioxide with the boundary condition of constant wall heat flux. Under these conditions the density and thermal properties vary greatly with temperature. The results indicate that unusual behavior of the heat-transfer coefficient may occur for certain values of heat flux and fluid temperature. The new solution technique presented should be applicable to a variety of partial differential equations with variable coefficients.