Laminar Bingham fluid flow in the entrance region of a circular pipe is investigated using a momentum integral model. The fully developed flow is uniform in the core region, while the velocity changes in the annular part of the cross section of the pipe. The inlet-filled region concept is adopted. In the inlet region, the boundary layer thickness increases until the size of the plug flow area reaches the fully developed flow size. The model converges to the fully developed solution in the filled region. The model provides the velocity, pressure drop, and skin friction coefficient profiles. The pressure drop results are in good agreement with published experimental data. The flow results asymptotically converge to the fully developed values. In addition, the results are consistent with published Newtonian fluid flow experimental data and theoretical results for the boundary layer thickness, pressure drop, and centerline velocity for small values of the Bingham number.