Weak electron-phonon coupling in Au and Cu produces a significant thermal resistance when heat flows from a thin Pt layer into a thin Au or Cu layer on picosecond time scales. Metal bilayers (Pt/Au and Pt/Cu) were prepared by magnetron sputter deposition; thermal transport in the bilayers was studied by time domain thermoreflectance in the temperature range 38<T<300 K. Analysis of heat transfer in the bilayer yields the electron-phonon coupling parameter g(T) of Au and Cu. Our results for g(T) are consistent with the temperature dependence predicted by the two-temperature model of Kaganov et al. [Sov. Phys. JETP 4, 173 (1957)] and help bridge the gap between data obtained using pump-probe spectroscopy at room temperature and electrical measurements at low temperatures.