  We construct intersecting brane configurations in Anti-de-Sitter space
localizing gravity to the intersection region, with any number $n$ of extra
dimensions. This allows us to construct two kinds of theories with infinitely
large new dimensions, TeV scale quantum gravity and sub-millimeter deviations
from Newton's Law. The effective 4D Planck scale $M_{Pl}$ is determined in
terms of the fundamental Planck scale $M_*$ and the $AdS$ radius of curvature
$L$ via the familiar relation $M_{Pl}^2 \sim M_{*}^{2+n} L^n$; $L$ acts as an
effective radius of compactification for gravity on the intersection. Taking
$M_* \sim $ TeV and $L \sim $ sub-mm reproduces the phenomenology of theories
with large extra dimensions. Alternately, taking $M_* \sim L^{-1} \sim M_{Pl}$,
and placing our 3-brane a distance $\sim 100 M_{Pl}^{-1}$ away from the
intersection gives us a theory with an exponential determination of the
Weak/Planck hierarchy.
