Heat transfer for hydrodynamically and thermally fully developed laminar flow in a curved pipe is solved by a method of series expansion. The wall temperature around the periphery of any cross section, the mean heat flux along the pipe, and the internal heat generation are assumed to be uniform. The first few terms of the series expansion are determined and analytical expressions for the temperature distribution in the fluid and for the Nusselt number are presented; the results are applicable for small curvatures. A correlation with experiment is included.