  In theories with (sets of) two large extra dimensions and supersymmetry in
the bulk, the presence of non-supersymmetric brane defects naturally induces a
logarithmic potential for the volume of the transverse dimensions. Since the
logarithm of the volume rather than the volume itself is the natural variable,
parameters of O(10) in the potential can generate an exponentially large size
for the extra dimensions. This provides a true solution to the hierarchy
problem, on the same footing as technicolor or dynamical supersymmetry
breaking. The area moduli have a Compton wavelength of about a millimeter and
mediate Yukawa interactions with gravitational strength. We present a simple
explicit example of this idea which generates two exponentially large
dimensions. In this model, the area modulus mass is in the millimeter range
even for six dimensional Planck scales as high as 100 TeV.
