In this paper, we consider a linear two-stage robust optimization model with a mixed integer recourse problem. Currently, this type of two-stage robust optimization model does not have any exact solution algorithm available. We first present a set of sufficient conditions under which the existence of an optimal solution is guaranteed. Then, we present a nested column-and-constraint generation algorithm that can derive an exact solution in nite steps. The algorithm development also yields novel solution methods to solve general mixed integer bi-level programs and some four-level programs. Finally, the proposed framework is demonstrated through an application of a robust rostering problem with part-time staff scheduling in the second stage.