A polyatomic gas with slow relaxation of the internal modes is considered, and the Navier-Stokes equations with two temperatures, the translational and internal temperatures, are derived for such a gas on the basis of the ellipsoidal-statistical (ES) model of the Boltzmann equation for a polyatomic gas, proposed by Andries et al. [Eur. J. Mech. B, Fluids 19, 813 (2000)10.1016/S0997-7546(00)01103-1], by the Chapman-Enskog procedure. Then, the derived equations are applied to numerically investigate the structure of a plane shock wave in CO_{2} gas, which is known to have slowly relaxing internal modes. The results show good agreement with those obtained by the direct numerical analysis of the ES model for moderately strong shock waves. In particular, the results perfectly reproduce the double-layer structure of the shock profiles consisting of a thin front layer with rapid change and a thick rear layer with slow relaxation of the internal modes.