High wind energy penetration critically challenges the economic dispatch of current and future power systems. Supply and demand must be balanced at every bus of the grid, while respecting transmission line ratings and accounting for the stochastic nature of renewable energy sources. Aligned to that goal, a network-constrained economic dispatch is developed in this paper. To account for the uncertainty of renewable energy forecasts, wind farm schedules are determined so that they can be delivered over the transmission network with a prescribed probability. Given that the distribution of wind power forecasts is rarely known, and/or uncertainties may yield non-convex feasible sets for the power schedules, a scenario approximation technique using Monte Carlo sampling is pursued. Upon utilizing the structure of the DC optimal power flow (OPF), a distribution-free convex problem formulation is derived whose complexity scales well with the wind forecast sample size. The efficacy of this novel approach is evaluated over the IEEE 30-bus power grid benchmark after including real operation data from seven wind farms.