  We consider the zero heat conductivity limit to a contact discontinuity for
the mono-dimensional full compressible Navier-Stokes-Fourier system. The method
is based on the relative entropy method, and do not assume any smallness
conditions on the discontinuity, nor on the $BV$ norm of the initial data. It
is proved that for any viscosity $\nu\geq0$, the solution of the compressible
Navier-Stokes-Fourier system (with well prepared initial value) converges, when
the heat conductivity $\kappa$ tends to zero, to the contact discontinuity
solution to the corresponding Euler system. We obtain the decay rate
$\kappa^{\frac12}$. It implies that the heat conductivity dominates the
dissipation in the regime of the limit to a contact discontinuity. This is the
first result, based on the relative entropy, of an asymptotic limit to a
discontinuous solutions for a system.
