An experimental setup to measure the thermal contact conductance across a silicon-copper (Si-Cu) interface is described, and the results obtained are presented. The resulting thermal contact resistance data are used in estimating the thermo-mechanical and optical performance of optical substrates cooled by interfaced copper cooling blocks. Several factors influence the heat transfer across solid interfaces. These include the material properties, interface pressure, flatness and roughness of the contacting surfaces, temperature, and interstitial material, if any. Results presented show the variation of thermal contact conductance as a function of applied interface pressure for a Cu-Si interface. Various interstitial materials investigated include indium foil, silver foil and a liquid eutectic (Ga-In-Sn). As expected, thermal contact resistance decreases as interface pressure increases, except in the case of the eutectic, in which it was nearly constant. The softer the interstitial material, the lower the thermal contact resistance. Liquid metal provides the lowest thermal contact resistance across the Cu-Si interface, followed by the indium foil, and then the silver foil.