We consider a Helmholtz transmission problem with stochastic interface. For this case, the map from the high-dimensional, uncertain parameter to point values of the solution on the physical domain is non-smooth, posing a challenge for many methods to compute surrogates of this map. However, the mapping is piecewise-smooth, and we exploit this to justify theoretically why deep neural networks with ReLU activation function can provide good surrogates. We also show numerical results illustrating their good performance in practice for low-frequency regimes. The experiments also evidence a deterioration of the neural network performance as the wavenumber increases. We discuss the reason for this and possible remedies.