<p>This paper is concerned with the vanishing dissipation limiting problem of one-dimensional non-isentropic Navier-Stokes equations with shock data. The limiting problem was solved by Hoff and Liu (1989) for isentropic gas with a single shock, but was left open for the non-isentropic case. In this paper, we solve the non-isentropic case, i.e., we first establish the global existence of solutions to the non-isentropic Navier-Stokes equations with initial discontinuous shock data, and then show these solutions converge in the <i>L</i>^∞ norm to a single shock wave of the corresponding Euler equations away from the shock curve in any finite time interval, as both the viscosity and heat-conductivity tend to zero. Different from Hoff and Liu’s work in which an integrated system was essentially used, we introduce a time-dependent shift <b>X</b>^<i>ε</i>(<i>t</i>) to the viscous shock so that a weighted Poincaré inequality can be applied to overcoming the difficulty generated from the “bad” sign of the derivative of the viscous shock velocity, and the anti-derivative technique is not needed. We also obtain an intrinsic property of the non-isentropic viscous shock (see Lemma 2.3 below). With the help of Lemma 2.3, we can derive the desired uniform <i>a priori</i> estimates of solutions, which can be shown to converge in the <i>L</i>^∞ norm to a single inviscid shock in any given finite time interval away from the shock, as the vanishing dissipation limit. Moreover, the shift <b>X</b>^<i>ε</i>(<i>t</i>) tends to zero in any finite time as viscosity tends to zero. The proof consists of a scaling argument, the <i>L</i>²-contraction technique with a time-dependent shift to the shock, and the relative entropy method.</p>