We study the asymmetric six-vertex model in the quadrant with parameters on the stochastic line. We show that the random height function of the model converges to an explicit deterministic limit shape as the mesh size tends to $0$. We further prove that the one-point fluctuations around the limit shape are asymptotically governed by the GUE Tracy–Widom distribution. We also explain an equivalent formulation of our model as an interacting particle system, which can be viewed as a discrete time generalization of ASEP started from the step initial condition. Our results confirm a 1992 prediction of Gwa and Spohn that this system belongs to the KPZ universality class.