We develop a sequential sampling procedure for solving a class of stochastic programs. A sequence of feasible solutions, with at least one optimal limit point, is given as input to our procedure. Our procedure estimates the optimality gap of a candidate solution from this sequence, and if that point estimate is sufficiently small then we stop. Otherwise, we repeat with the next candidate solution from the sequence with a larger sample size. We provide conditions under which this procedure: (i) terminates with probability one and (ii) terminates with a solution which has a small optimality gap with a prespecified probability.