We provide stabilizing feedback control laws for the kinematic system of an underactuated axisymmetric spacecraft subject to input constraints, The proposed control law forces all closed-loop trajectories in a region of the state space where the control inputs are small and bounded. The control law is subsequently extended to solve the case of attitude tracking for an underactuated spacecraft using two controls. As a special case we also provide a feedback control for the spacecraft symmetry axis to track a specified direction in the inertial space, All proposed control laws achieve asymptotic stability with exponential convergence. In addition, we give a methodology for generating feasible trajectories using the fact that the system is differentially flat. One of the novelties of the proposed control design is the use of a non-standard description of the attitude motion, which allows the decomposition of the general motion into two rotations. This attitude description is especially useful for analyzing axisymmetric bodies, where the motion of the symmetry axis is of prime importance.