In this paper, an angular velocity-free attitude control system is designed for an orbiting three-axis stabilized satellite with flexible appendages, taking into account parametric uncertainties and external disturbances (constant and periodic ones). For the controller designed, it incorporates a low-pass linear filter that is derived without explicit differentiation of attitude to synthesize angular velocity-like signals. Moreover, an integral feedback of the attitude was explicitly included to compensate the constant disturbances. The associated stability proof is constructive and accomplished by the development of a Lyapunov function candidate, which contains cross states and elastic vibration variables. The uniformly bounded stability of the closed-loop system is guaranteed and the elastic vibration was also passively reduced provided that appropriate robustness conditions on the controller gains are satisfied. These conditions use partial knowledge of the parameters describing the elastic dynamics. The closed-loop performance of the control solution derived here is evaluated extensively through numerical simulations.