A very precise low-energy electron diffractometer capable of operating from near liquid He temperatures to near 1000 K has been developed. As a first experiment, the Debye-Waller factor and the thermal diffuse scattering from a Ag(111) crystal have been studied. In earlier experiments, owing to the large multiphonon scattering, the one-phonon scattering could be determined only in the central part of the Brillouin zone. At low temperatures the one-phonon scattering is relatively much larger. Hence, using their known temperature dependences in the high temperature limit, the zero-, one-, and multiphonon contributions have been separated all the way to the zone boundary. Near the zone boundary the deviation from a continuum model due to the phonon dispersion in a discrete lattice is observed, and the angular, energy penetration, and temperature dependences have been observed.