We prove the stability of a planar contact discontinuity without shear, a family of special discontinuous solutions for the three-dimensional full Euler system, in the class of vanishing dissipation limits of the corresponding Navier-Stokes-Fourier system. We also show that solutions of the Navier-Stokes-Fourier system converge to the planar contact discontinuity when the initial datum converges to the contact discontinuity itself. This implies the uniqueness of the planar contact discontinuity in the class that we are considering. Our results give an answer to the open question, whether the planar contact discontinuity is unique for the multi-D compressible Euler system. Our proof is based on the relative entropy method, together with the theory of a-contraction up to a shift and our new observations on the planar contact discontinuity.