  The spatial distribution of persistent spins at zero-temperature in the pure
two-dimensional Ising model is investigated numerically. A persistence
correlation length, $\xi (t)\sim t^Z$ is identified such that for length scales
$r<<\xi (t)$ the persistent spins form a fractal with dimension $d_f$; for
length scales $r>>\xi (t)$ the distribution of persistent spins is homogeneous.
The zero-temperature persistence exponent, $\theta$, is found to satisfy the
scaling relation $\theta = Z(2-d_f)$ with $\theta =0.209\pm 0.002, Z=1/2$ and
$d_f\sim 1.58$.
