In this paper, we present a conservative finite volume scheme for the gas dynamics in Lagrangian coordinates, which is fast and nondiffusive. Fast, because it relies on an approximate Riemann solver, and hence the costly resolution of Riemann problems is avoided. Nondiffusive, because the solution is exact when the initial data is an admissible isolated shock, and discontinuities are sharply captured in general. The construction of the scheme uses two main tools: the extension to the barotropic Euler equations of the discontinuous reconstruction strategy presented in [Agu14], and the approximate Riemann solver of [CC13], which is exact on isolated admissible shocks. Numerical experiments in 1D and 2D are proposed.