Heat transport across interfaces is a ubiquitous phenomenon with many unresolved aspects. In particular, it is unknown if an interfacial thermal resistance (ITR) occurs in matter with high-energy-density where free electrons dominate the heat conduction. Here, we report on the first experimental evidence that a significant heat barrier is present between two different regions of high-energy-density matter: a strongly heated tungsten wire and a surrounding plastic layer that stays relatively cold. We use diffraction-enhanced imaging to track the time evolution of density discontinuities and reconstruct the temperature evolution in the quasi-stationary stage. The clear signatures of a temperature jump demonstrate the importance of the ITR for strongly heated systems with far-reaching implications for interpreting experiments and applications like inertial confinement fusion.