Fully developed laminar flow through a heated horizontal curved tube is studied using a two-dimensional numerical representation of the fully-elliptic Navier-Stokes and energy equations. Buoyancy and curvature effects are included and two types of thermal boundary conditions are examined. Heat transfer and pressure drop data are provided in addition to flow velocity and temperature contours. Regime maps are presented which delineate the range of combined curvature/buoyancy influence for both types of boundary conditions. The results for uniform peripheral heat flux reveal a large region of parameter space where fully developed laminar solutions could not be found.