  We present strong numerical evidence for the existence of a
localization-delocalization transition in the eigenstates of the 1-D Anderson
model with long-range hierarchical hopping. Hierarchical models are important
because of the well-known mapping between their phases and those of models with
short range hopping in higher dimensions, and also because the renormalization
group can be applied exactly without the approximations that generally are
required in other models. In the hierarchical Anderson model we find a finite
critical disorder strength Wc where the average inverse participation ratio
goes to zero; at small disorder W < Wc the model lies in a delocalized phase.
This result is based on numerical calculation of the inverse participation
ratio in the infinite volume limit using an exact renormalization group
approach facilitated by the model's hierarchical structure. Our results are
consistent with the presence of an Anderson transition in short-range models
with D > 2 dimensions, which was predicted using renormalization group
arguments. Our finding should stimulate interest in the hierarchical Anderson
model as a simplified and tractable model of the Anderson localization
transition which occurs in finite-dimensional systems with short-range hopping.
