This paper presents an efficient second-order finite volume method for the simulation of bedload sediment transport which is capable of modeling wet/dry fronts. The governing equations comprise the shallow water equations (SWEs) for the hydrodynamic phase and the Exner equation for the estimation of bedload sediment transport. These sets of equations are then solved using a weakly coupled scheme based on an augmented Riemann solver (WCAR). In this approach, first, the morphodynamic equation is solved, and then, updated bedload changes with the same Riemann structures are used as a source term within the SWEs. The Smart formula is utilized to estimate the bedload sediment discharge within the Exner equation. The proposed numerical model is first used to model a parabolic sediment layer. Then, a bedload hump propagation with an initial subcritical condition is considered. Next, the simulation of dam-break flow over a mobile bed is investigated. Finally, the dam failure due to over-topping is studied and the computed results are compared with available experimental data. Numerical results indicate that the introduced weakly coupled method, developed based upon the augmented Riemann solver, can be effectively used for modeling all investigated flow regimes, including dry-state interfaces.