This paper develops criteria for designing interval observers to guarantee robustness with respect to disturbances for feedback control of nonlinear systems. Intervals in which components of the state vector are guaranteed to stay are estimated based on the information of the range of the initial state and the disturbances. For formulating desirable properties of boundedness and convergence of estimated intervals in the presence of disturbances, the notion of integral input-to-state stability is introduced to interval observer design. Guaranteed properties of the observer-based feedback designed in the formulation are not only global, but also address nonlinearities which are broader than those covered by previously existing approaches.