In this talk, I present an exact solution method for two-stage stochastic programs under decision-dependent uncertainty. In such problems, first-stage decisions determine the probability distribution of second-stage uncertain parameters.  Particularly, we focus on a broad class of problems where the number of potential probability distributions is finite but exponentially large. In these problems, the recourse function is non-convex and discontinuous even for problems with continuous second-stage. The proposed method extends the well-known L-Shaped method and is applicable also to two-stage stochastic programs with integer variables at both stages. We show that the algorithm converges finitely and propose a number additional improvements that facilitate convergence, such as valid inequalities. Results of numerical experiments on a facility location problems under endogenous uncertainty show promising scalability and efficiency also on very challenging problems.