We present a model and an exact solution method for single-source capacitated facility location problems with decision-dependent demand uncertainty. The model allows the joint probability distribution of customer demands to depend on the facility to which demand points are allocated. The number of possible decision-dependent distributions is therefore equal to the number of feasible allocations of demand points to (groups of) facilities. To address the exponential size of the model, we apply and extend a recent version of the L-Shaped method for stochastic programs with decision-dependent uncertainty. In particular, we compare the original optimality cut with a new optimality cut based on a bilinear reformulation. We prove that, separated together with the original one, the new cut yields a tighter restriction, at the expense of introducing bilinear terms. In addition, we prove an effective decision-independent valid inequality that significantly improves convergence. Extensive numerical tests on 180 instances demonstrate that the method, supported by the valid inequality, can find optimal or near-optimal solutions to problem instances with up to $6^20$ potential probability distributions. The tests also show that, while the new cut closes slightly more instances, the original cut remains preferable in terms of solution time and optimality gap.