In this paper we consider the vehicle routing problem with hard time windows and stochastic service times (VRPTW-ST); in this variant of the classic VRPTW the service times are random variables. In particular, given a set of vehicle routes, some of the actual service times might not be feasible given the customer time windows. We consider a stochastic programming approach to formulate the VRPTW-ST and, to the best of our knowledge, we are the first to do so. We provide a new set partitioning formulation that includes a constraint on the minimum success probability of the set of vehicle routes. Under some mild conditions, we develop a method to exactly compute the success probability of the routes. We then solve the VRPTW-ST by a branch-price-and-cut algorithm, where the main challenges are in the solution of the subproblems of the column generation procedure. We adapt the dynamic programming algorithm to account for the probabilistic resource consumption by extending the label dimension and by providing new dominance rules. Extensive computational experiments prove the effectiveness of both the solution method and the stochastic model.