We consider a general class of two-stage distributionally robust optimization (DRO) problems where the ambiguity set is constrained by fixed marginal probability laws that are not necessarily discrete. We derive primal and dual formulations of this class of problems and subsequently develop a numerical algorithm for computing approximate optimizers as well as approximate worst-case probability measures. Moreover, our algorithm computes both an upper bound and a lower bound for the optimal value of the problem, where the difference between the computed bounds provides a direct sub-optimality estimate of the computed solution. Most importantly, the sub-optimality can be controlled to be arbitrarily close to 0 by appropriately choosing the inputs of the algorithm. To demonstrate the effectiveness of the proposed algorithm, we apply it to three prominent instances of two-stage DRO problems in task scheduling, multi-product assembly, and supply chain network design with edge failure. The ambiguity sets in these problem instances involve a large number of continuous or discrete marginals. The numerical results showcase that the proposed algorithm computes high-quality robust decisions along with non-conservative sub-optimality estimates.