  We study how the ground state of the two-dimensional Ising spin glass with
Gaussian interactions in zero magnetic field changes on altering the boundary
conditions. The probability that relative spin orientations change in a region
far from the boundary goes to zero with the (linear) size of the system L like
L^{-lambda}, where lambda = -0.70 +/- 0.08. We argue that lambda is equal to
d-d_f where d (=2) is the dimension of the system and d_f is the fractal
dimension of a domain wall induced by changes in the boundary conditions. Our
value for d_f is consistent with earlier estimates. These results show that, at
zero temperature, there is only a single pure state (plus the state with all
spins flipped) in agreement with the predictions of the droplet model.
