  It is proven rigorously that the ground state in the Edwards-Anderson spin
glass model is unique in any dimension for almost all continuous random
exchange interactions under a condition that a single spin breaks the global
${\mathbb Z}_2$ symmetry. This theorem implies that replica symmetry breaking
does not occur at zero temperature. The site- and bond-overlap are concentrated
at their maximal values. It is argued that behaviors of short range spin glass
models are much different from those of mean field spin glass models near zero
temperature. Errata have been attached at the final page.
