Implementing accurate Distribution System State Estimation (DSSE) faces several challenges, among which the lack of observability and the high density of the distribution system. While data-driven alternatives based on Machine Learning models could be a choice, they suffer in DSSE because of the lack of labeled data. In fact, measurements in the distribution system are often noisy, corrupted, and unavailable. To address these issues, we propose the Deep Statistical Solver for Distribution System State Estimation (DSS^2), a deep learning model based on graph neural networks (GNNs) that accounts for the network structure of the distribution system and the governing power flow equations of the problem. DSS^2 is based on GNN and leverages hypergraphs to model the network as a graph into the deep-learning algorithm and to represent the heterogeneous components of the distribution systems. A weakly supervised learning approach is put forth to train the DSS^2: by enforcing the GNN output into the power flow equations, we force the DSS^2 to respect the physics of the distribution system. This strategy enables learning from noisy measurements and alleviates the need for ideal labeled data. Extensive experiments with case studies on the IEEE 14-bus, 70-bus, and 179-bus networks showed the DSS^2 outperforms the conventional Weighted Least Squares algorithm in accuracy, convergence, and computational time while being more robust to noisy, erroneous, and missing measurements. The DSS^2 achieves a competing, yet lower, performance compared with the supervised models that rely on the unrealistic assumption of having all the true labels.