Transmission channeling and ion scattering studies show that during the initial adsorption of atomic deuterium on Cu(100), 0.5 ML (monolayer) of D chemisorbs in the fourfold hollow sites at the unreconstructed surface. Additional adsorption of 0.5 ML of D induces a (2 X 2)p4g reconstruction of Cu(100) for which Angstrom 0.23 A lateral displacement of the first-layer Cu atoms is measured. It is proposed that these latter 0.5 ML D atoms populate displaced tetrahedral subsurface sites, and that it is the subsurface D which drives the reconstruction.