This paper considers nonlinear semi-infinite problems, which contain at least one semi-infinite constraint (SIC). The standard branch-and-bound algorithm is adapted to such problems by extending usual upper and lower bounding techniques for nonlinear inequality constraints to SICs. This is achieved by defining the interval evaluation of parametrized functions and their generalized gradients, by also adapting numerical constraint programming techniques to quantified inequalities, and by introducing linear relaxations and restrictions for SICs. The overall efficiency of our algorithm is demonstrated on a standard set of benchmarks from the literature, in comparison with the best state of the art alternative.