This work addresses the determination of a zero-order term in a one-dimensional wave equation with a discontinuous (piecewise C-2 )main coefficient. The study establishes Lipschitz stability of the inverse problem without monotonicity assumptions on the jumps. Moreover, it is demonstrated that by imposing monotonicity on the jumps, typically required in the multidimensional setting, precise determination of the condition over a large time for Lipschitz stability becomes possible. We also show that the C-bRec algorithm proposed by L. Baudouin and collaborators in [2] is also theoretically well adapted to the discontinuous case. The proof is based on the BukhgeimKlibanov method combined with a one-parameter Carleman estimate for the one-dimensional wave equation with a discontinuous main coefficient.