This paper presents a sim-exact algorithm for solving a stochastic capacitated dispersion problem under uncertainty. The method extends previous simheuristic approaches by replacing metaheuristic search with an iterative sequence of mixed-integer programming models, combined with simulation to evaluate stochastic feasibility and solution quality. Capacity uncertainty is modeled using a scaled Beta distribution. A chance constraint ensures that total effective capacity meets a required threshold with a given reliability level. To handle the nonlinearity of the dispersion objective, a mixed-integer linear programming formulation based on a telescopic representation of pairwise distances is used. The sim-exact procedure solves a sequence of increasingly conservative models controlled by a safety stock parameter. An incumbent solution pool is maintained across iterations. This pool is used to generate warm starts, improve bounds, reuse feasible solutions, and define stopping criteria. These mechanisms allow information to be transferred between iterations and reduce computational effort. Computational experiments are conducted on standard benchmark instances and compared against a simheuristic algorithm and a sample average approximation framework. Results show that the sim-exact method consistently produces higher-quality solutions than the simheuristic approach, with an average improvement of around 9.8% in objective value. It also yields stronger solutions than sample average approximation, which shows limited reliability in producing feasible solutions even for a large number of scenarios.