Fully-developed, laminar, steady, free-and-forced convection heat transfer in an electrically-conducting fluid flowing in an electrically-insulated, horizontal, circular pipe (with its wall subjected to a uniform heat flux) in a vertical, uniform, transverse magnetic field was solved numerically using several finite difference schemes for Grashof numbers from 0 to 10(6) and Hartmann numbers from 0 to 500. For high Hartmann numbers, a refinement of the mesh in the radial direction was necessary in the Hartmann boundary layer and relaxation techniques were needed to have a convergent solution. Velocity profiles, temperature profiles and local and average Nusselt numbers are reported for combined free-and-forced convection.