We show that the persistence probability [Formula Presented] in a coarsening system of linear size L at a time t, has the finite-size scaling form [Formula Presented] where θ is the persistence exponent and z is the coarsening exponent. The scaling function [Formula Presented] for [Formula Presented] and is constant for large x. The scaling form implies a fractal distribution of persistent sites with power-law spatial correlations. We study the scaling numerically for the Glauber-Ising model at dimension [Formula Presented] to 4 and extend the study to the diffusion problem. Our finite-size scaling ansatz is satisfied in all these cases providing a good estimate of the exponent [Formula Presented]. © 2000 The American Physical Society.