  We study the nature of the phase transition in the multifractal formalism of
the harmonic measure of Diffusion Limited Aggregates (DLA). Contrary to
previous work that relied on random walk simulations or ad-hoc models to
estimate the low probability events of deep fjord penetration, we employ the
method of iterated conformal maps to obtain an accurate computation of the
probability of the rarest events. We resolve probabilities as small as
$10^{-70}$. We show that the generalized dimensions $D_q$ are infinite for
$q<q^*$, where $q^*= -0.17\pm 0.02$. In the language of $f(\alpha)$ this means
that $\alpha_{max}$ is finite. We present a converged $f(\alpha)$ curve.
