We build a non-dissipative second order algorithm for the approximate resolution of the one-dimensional Euler system of compressible gas dynamics with two components. The considered model was proposed in [Allaire-Clerc-Kokh 2002]. The algorithm is based on [Kokh-Lagoutière 2010] which deals with a non-dissipative first order resolution in Lagrange-remap formalism. In the present paper we describe, in the same framework, an algorithm that is second order accurate in time and space, and that preserves sharp interfaces. Numerical results reported at the end of the paper are very encouraging, showing the interest of the second order accuracy for genuinely non-linear waves.