Estimating river discharge from in situ and/or remote sensing data is a key issue for evaluation of water balance at local and global scales and for water management. Variational data assimilation (DA) is a powerful approach used in operational weather and ocean forecasting, which can also be used in this context. A distinctive feature of the river discharge estimation problem is the likely presence of significant uncertainty in principal parameters of a hydraulic model, such as bathymetry and friction, which have to be included into the control vector alongside the discharge. However, the conventional variational DA method being used for solving such extended problems often fails. This happens because the control vector iterates (i.e., approximations arising in the course of minimization) result into hydraulic states not supported by the model. In this paper, we suggest a novel version of the variational DA method specially designed for solving estimation-under-uncertainty problems, which is based on the ideas of iterative regularization. The method is implemented with SIC2, which is a full Saint-Venant based 1D-network model. The SIC2 software is widely used by research, consultant and industrial communities for modeling river, irrigation canal, and drainage network behavior. The adjoint model required for variational DA is obtained by means of automatic differentiation. This is likely to be the first stable consistent adjoint of the 1D-network model of a commercial status in existence. The DA problems considered in this paper are offtake/tributary estimation under uncertainty in the cross-device parameters and inflow discharge estimation under uncertainty in the bathymetry defining parameters and the friction coefficient. Numerical tests have been designed to understand identifiability of discharge given uncertainty in bathymetry and friction. The developed methodology, and software seems useful in the context of the future Surface Water and Ocean Topography satellite mission.