This paper presents a fully distributed algorithm for the stochastic Volt/VAr control (VVC) problem in active distribution networks. Exact convexification of the VVC problem is achieved through the use of second-order cones on the continuous relaxation of the original optimization structure. The global optimum solution of the relaxed problem is obtained through the application of the alternating direction method of multipliers (ADMM). In order to minimize the effect of rounding off on the final solution, an adaptive threshold discretization technique is used. A two-stage control strategy is adopted where the discrete controllers, like load tap changers and switched capacitors, are dispatched at the beginning of the optimization interval and the continuous controllers, like distributed generation (DG) inverters, are adjusted in real time according to an optimized decision rule. The superiority of the proposed algorithm is demonstrated through numerical simulations on the UKGDS-95 and the IEEE-123 bus systems.