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We report on a novel approach to measure the phase relaxation length and femtosecond lifetime of hot quasiparticles on metal surfaces. A 4 K scanning tunneling microscope has been used to study the spatial decay of interference patterns in the local density of states for surface state electrons on Ag(111) and Cu(111). This decay is governed by inelastic; electron-electron scattering. We find a (E - E-F)(-2) energy dependence of the lifetimes for both Ag and Cu, and our values are comparable to the corresponding bulk electron lifetimes. This indicates that electron-electron interaction of hot surface state electrons with the Fermi sea is dominated by the underlying bulk electrons. [S0031-9007(99)09260-1].