We have used pulsed photothermal radiometry to measure the thermal contact resistance at the interface of a smooth polymer film and a polished metal substrate. This method relies on the heating of the film surface by a short light pulse and detecting the subsequent infrared thermal radiation from the surface. An analytical solution to the heat diffusion equation shows that in a suitable delayed time interval, the infrared signal decays exponentially in time with a time constant related to the thermal contact resistance of the interface. By changing gases in the interface at constant pressures, we are able to separate the thermal conductance into two components: that due to solid contacts and that due to gas conduction. The thermal conductance due to gas conduction in the interface is proportional to the thermal conductivity of the gas found in continuum fluid theory, except for He which is more than 30% lower. We believe that the discrepancy in He is partly due to the fact that the mean free path of He gas molecules is not much smaller than the mean width of the gaps in the interface and so the classical continuum fluid theory for heat conduction would no longer hold.