We show that DLA follows a surprising new scaling rule. It expresses that the screened region, in which the harmonic measure is tiny, increases more than proportionately as the cluster grows. This scaling rule also gives indirect evidence that the harmonic measure of lattice DLA follows a hyperbolic probability distribution of exponent equal to 1. This distribution predicts that sample moments behave erratically, hence explains why the common restricted multifractal formalism fails to apply to DLA. © 1991.