The temperature-dependent damping of quantum-mechanical interference patterns from surface-state electrons scattering off steps on Ag(111) and Cu(111) has been studied using scanning tunneling microscopy (STM) and spectroscopy in the temperature range 3,5-178 K. The thermal damping of the electron standing waves is described quantitatively within a simple plane-wave model accounting for thermal broadening due to the broadening of the Fermi-Dirac distributions of sample and tip, for beating effects between electrons with different kll vectors, and for inelastic collisions of the electrons, e.g., with phonons. Our measurements reveal that Fermi-Dirac broadening fully explains the observed damping for Ag and Cu. From the analysis of our data, lower limits of the phase-relaxation lengths at the Fermi energy EF Of the two-dimensional electron gas of L-phi(E-F)greater than or similar to 600 Angstrom at 3.5 K and greater than or similar to 250 Angstrom at 77 K for Ag(111), and of L-phi(E-F)greater than or similar to 660 Angstrom at 77 K and greater than or similar to 160 Angstrom at 178 K for Cu(111) are deduced. In contrast to integral measurements such as photoemission we measure L-phi close to EF and also locally. The latter eliminates residual line widths due to surface defect scattering found in the integrating techniques. Our STM results, therefore, currently provide a very good absolute estimate of L-phi and the inelastic lifetime tau=L-phi/v(F), respectively. Our values can be combined with photoemission results on dL(phi)/dT to derive the inelastic lifetime of surface state electrons at any T.