We present a many-body calculation for the conductance of a conducting bridge of a simple hydrogen molecule between Pt electrodes. The experimental results showed that the conductance G=dI/dV has the maximum value near the quantum unit G0=2e2/h. The I- V dependence presents peak and dip and we consider that the electron-phonon interaction is responsible for this behavior. At T=0 there is a step in this dependence for the energy of phonons ω0 which satisfies eV=ω0. We calculated the conductance at finite temperature and showed that dG(T)/dV∝1/4T cosh2 (eV-ω0)/2T.