The dual quaternion was used to investigate the 6-DOF (six degree-of-freedom) relative motion of spacecraft, and an integrated controller of attitude and orbit with input saturation constraint was designed. Firstly, the dual number and dual quaternion followed by the derivation of 6-DOF relative motion of attitude and orbit coupled model were introduced. Then, an explicitly bounded controller was proposed by using the property that the absolute value of hyperbolic tangent function was less than 1. Two coupled adaptive regulators were respectively designed to adjust the outputs of the controller dynamically. Thirdly, it is proved by a rigorous theoretical analysis that the closedloop system is globally and asymptotically stable. Finally, numerical simulation experiment is implemented to demonstrate the validity and effectiveness of the proposed controller. Experimental results show that both the obtained control torque and control force satisfy the given constraints, the 6-DOF relative motion of spacecraft can be controlled accurately and the method is robust to parameter uncertainties and external disturbance. Compared with other methods, the proposed method has the following advantages: the convergence rate is faster, the max magnitude of output control torque and control force are both smaller, and the consumed energy is less.