This paper presents a Lagrangian cell-centered conservative gas dynamics scheme.The piecewise constant pressures of cells arising from the current time sub-cell densities and the current time isentropic speed of sound are introduced.Multipling the initial cell density by the initial sub-cell volumes obtains the sub-cell Lagrangian masses,and dividing the masses by the current time sub-cell volumes gets the current time subcell densities.By the current time piecewise constant pressures of cells,a scheme that conserves the momentum and total energy is constructed.The vertex velocities and the numerical fluxes through the cell interfaces are computed in a consistent manner due to an original solver located at the nodes.The numerical tests are presented,which are representative for compressible flows and demonstrate the robustness and accuracy of the Lagrangian cell-centered conservative scheme.