We present a new first order numerical method on lagrangian Voronoï moving meshes for the numerical simulation of compressible flows with shocks and internal interfaces between different gas. The method is based on the closed form formula of the partial derivative of the volume of Voronoï cells with respect to the generators. The mathematical proof of the formula seems original with respect to the literature. A corollary is that the volume of Voronoï cells is generically of class C 1 with respect to the generators. The final scheme is conservative in local mass, total momentum and total energy, and it is endowed with an entropy inequality which insures the correctness of shocks calculations. Numerical illustrations in dimension d = 2 are displayed for basic problems on coarse meshes. The implementation developed to obtain the numerical illustrations uses a freely available library for the generation of the Voronoï cells at all time steps.