The method of iterated conformal maps allows one to study the harmonic measure of diffusion-limited aggregates with unprecedented accuracy. We employ this method to explore the multifractal properties of the measure, including the scaling of the measure in the deepest fjords that were hitherto screened away from any numerical probing. We resolve probabilities as small as [formula presented] and present an accurate determination of the generalized dimensions and the spectrum of singularities. We show that the generalized dimensions [formula presented] are infinite for [formula presented] where [formula presented] is of the order of [formula presented] In the language of [formula presented] this means that [formula presented] is finite. The [formula presented] curve loses analyticity (the phenomenon of “phase transition”) at [formula presented] and a finite value of [formula presented] We consider the geometric structure of the regions that support the lowest parts of the harmonic measure, and thus offer an explanation for the phase transition, rationalizing the value of [formula presented] and [formula presented] We thus offer a satisfactory physical picture of the scaling properties of this multifractal measure.