The developing flow and heat transfer in the entry region of a heated circular tube is analyzed for the case of constant wall temperature. An integral or boundary-layer solution is presented which has a number of advantages over earlier Karman-Pohlhausen integral analyses. Thus, in the present analysis, the velocity and temperature distributions, the local and mean drag coefficients, and the local and mean Nusselt numbers approach their fully-developed values asymptotically. The new analysis is based on the hydrodynamic inlet-filled region concept originally proposed by Ishizawa (1966) and later adopted by Mohanty and Asthana (1978) to flow through a circular tube. This concept is extended to the combined entry-length problem by introducing a thermal transition region, herein called the thermally-filled region, between the thermal inlet boundary-layer region and the thermally fully-developed region. A thermal shape factor is also introduced which ensures smooth transition of all pertinent thermal quantities from the entrance region to the fully-developed region. Results for the Variation of the local and mean Nusselt numbers with axial distance along the tube for Pr = 0.1, 0.5, 0.7, 1, 5, and 10 are presented. These results agree well with the numerical solutions of Hornbeck (1965), Manohar (1969), and Hwang and Sheu (1974) and also with the correlations of Churchill and Ozoe (1973).