Solving the Shallow-Water-Sensitivity Equations for discontinuous flows involves the discretization of a Dirac source term accounting for disconti- nuities. Failing to account for this source term usually results in solution instability, with empirical sensitivity solutions exhibiting artificial peaks in the neighbourhood of shocks. An extension of the Harten-Lax-van Leer approximate Riemann solver is presented that allows the one-dimensional shallow-water-sensitivity equations to be discretized more accurately than in previously published versions. A discretization of boundary conditions and source terms is also provided. The proposed discretization allows for discontinuities in both the hydraulic and sensitivity boundary conditions. Numerical experiments indicate the superiority of the proposed approach for sensitivity analysis over the classical, empirical approach if the flow solution is discontinuous. A numerical convergence analysis demonstrates that the numerical and analytical solutions converge.