The aim of this thesis is to find a technique that allows for the use of decomposition methods known from stochastic programming in the framework of linear stochastic bilevel problems. The uncertainty is modeled as a discrete, finite distribution on some probability space. Two approaches are made, one using the optimal value function of the lower level, whereas the second technique uses the Karush-Kuhn-Tucker conditions of the lower level. Using the latter approach, an integer-programming based algorithm for the global resolution of these problems is presented and evaluated.