A theoretical model is presented for describing the motion of a deformable cell encapsulating a Newtonian fluid and enclosed by an elastic membrane in tube flow. In the mathematical formulation, the interior and exterior hydrodynamics are coupled with the membrane mechanics by means of surface equilibrium equations, and the problem is formulated as a system of integral equations for the interfacial velocity, the disturbance tube-wall traction, and the pressure difference across the two ends to the tube due to the presence of the cell. Numerical solutions obtained by a boundary-element method are presented for flow in a cylindrical tube with a circular cross-section, cytoplasm viscosity equal to the ambient fluid viscosity, and cells positioned sufficiently far from the tube wall so that strong lubrication forces do not arise. In the numerical simulations, cells with spherical, oblate ellipsoidal, and biconcave unstressed shapes enclosed by membranes that obey a neo-Hookean constitutive equation are considered. Spherical cells are found to slowly migrate toward the tube centerline at a rate that depends on the mean flow velocity, whereas oblate and biconcave cells are found to develop parachute and slipper-like shapes, respectively, from axisymmetric and more general initial orientations.