We consider a one-dimensional Saint-Venant system with uncertainty, for which we derive and analyze a novel stochastic Galerkin method. The proposed method is based on the truncated generalized polynomial chaos (gPC) expansion, whose coefficients satisfy a time-dependent system of PDEs, which is hyperbolic provided the water depth remains nonnegative at all times and for all values of both spatial and stochastic variables. We numerically solve the resulting system using a Riemann-problem-solver-free well-balanced and positivity-preserving finite-volume central-upwind scheme. The novelty of our approach is in the way we enforce the positivity of the computed water depth—this is achieved by generalizing the “draining time-step” technique to the system of gPC coefficients. We illustrate the performance of the proposed method on a number of challenging numerical examples. Though in some of the considered benchmarks, we obtain accurate mean and standard deviation of the stochastic solution, we realize that (small) oscillations appearing near the discontinuities propagate into the stochastic field and cause quite significant oscillations attributed to the Gibbs phenomenon. This demonstrates the limitations in the applicability of the stochastic Galerkin method to the problems with discontinuous solutions. As a possible way to remove (reduce) the aforementioned Gibbs oscillations, we propose to add an adaptive artificial viscosity to the system of gPC coefficients. However, this, like other existing filtering alternatives, affects the high resolution of the gPC stochastic Galerkin method.