A solution method for steady heat conduction in convectively cooled solids having temperature-dependent thermal conductivity is described and applied. The method provides solutions in algebraic form from which the temperature and its spatial derivatives can be evaluated at any point in the solid. To illustrate the application of the method, consideration is given to an internally heat-generating solid pierced by circular cooling passages. Results for the hot spot temperature are presented for a range of geometrical and thermal parameters. The results indicate a substantial influence of variable thermal conductivity when the temperature variation across the solid is large.