In this paper we propose an efficient approach for globally solving a class of convex semiinfinite programming (SIP) problems. Under the objective function and constraints (w.r.t. the variables to be optimized) convexity assumption, and appropriate differentiability, we propose a branch and bound exchange type method for SIP. To compute a feasible point for a SIP problem (and check feasibility) we need to solve a global optimization (sub-)problem, which is herein addressed by a branch and bound strategy. The major novelty of the proposed method consists in generating a sequence of feasible points for the SIP problem, obtained by a convex combination of a feasible point and the solution of a discretized finite optimization problem. A branch and bound strategy is also used to address the problem of minimizing the objective function, since we naturally obtain, as a result of the iterative process, bounds for the objective function. Under mild assumptions we prove convergence of the proposed algorithm. To illustrate the proposed approach, we provide some numerical results using some benchmark test problems.