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We establish existence of global-in-time weak solutions to the onedimensional, compressible Navier-Stokes system for a viscous and heatconducting ideal polytropic gas (pressure $p=K\theta/\tau$, internal energy$e=c_v \theta$), when the viscosity $\mu$ is constant and the heat conductivity$\kappa$ depends on the temperature $\theta$ according to $\kappa(\theta) =\bar \kappa \theta^\beta$, with $0\leq\beta<{3/2}$. This choice of degeneratetransport coefficients is motivated by the kinetic theory of gasses.  Approximate solutions are generated by a semi-discrete finite element scheme.We first formulate sufficient conditions that guarantee convergence to a weaksolution. The convergence proof relies on weak compactness and convexity, andit applies to the more general constitutive relations $\mu(\theta) = \bar \mu\theta^\alpha$, $\kappa(\theta) = \bar \kappa \theta^\beta$, with $\alpha\geq0$, $0 \leq \beta < 2$ ($\bar \mu, \bar \kappa$ constants). We then verify thesufficient conditions in the case $\alpha=0$ and $0\leq\beta<{3/2}$. The dataare assumed to be without vacuum, mass concentrations, or vanishingtemperatures, and the same holds for the weak solutions.

1D compressible flow with temperature dependent transport coefficients