This work proposes an exact solution approach for the two-machine robust flow shop problem, where operation processing times are uncertain and vary in a given interval. Based on the concept of budgeted uncertainty, the objective is to obtain a robust scheduling that minimizes the makespan of the restricted worst-case scenario, where only a subset of job processing times will oscillate to their worst-case values. To our knowledge, this is the first work to obtain optimal solutions to this problem, by extending two classical MILP formulations for the deterministic case and combining them with a Column-and-Constraint Generation framework. For this purpose, a dynamic programming algorithm was also developed, allowing the identification of worst-case scenarios in polynomial time. Experiments on a set of literature instances confirm the efficacy of our approach, including a case study that shows little overhead in the expected solution value of obtained robust solutions.