  We consider the Edwards-Anderson Ising spin glass model on the half-plane $Z
\times Z^+$ with zero external field and a wide range of choices, including
mean zero Gaussian, for the common distribution of the collection J of i.i.d.
nearest neighbor couplings. The infinite-volume joint distribution
$K(J,\alpha)$ of couplings J and ground state pairs $\alpha$ with periodic
(respectively, free) boundary conditions in the horizontal (respectively,
vertical) coordinate is shown to exist without need for subsequence limits. Our
main result is that for almost every J, the conditional distribution
$K(\alpha|J)$ is supported on a single ground state pair.
