  We study a non-Riemannian modification of 5-dimensional Kaluza-Klein theory.
In our proposal the Riemannian structure of the five-dimensional manifold is
replaced by a Weyl-integrable one. In this context a 4-dimensional
Poincar$\grave{e}$ invariant solution is studied. Spacetime structures with
thick smooth branes separated in the extra dimension arise. We focus our
attention, mainly, in a case where the massless graviton is located in one of
the thick branes at the origin, meanwhile the matter degrees of freedom are
confined to the other brane. Due to the small overlap of the graviton's
wave-function with the second thick brane, the model accounts for a resolution
of the mass hierarchy problem a la Randall-Sundrum. Although, initially, no
assumptions are made about the topology of the extra dimension, the solution
found yields an extra space that is effectivelly compact and respects $Z_2$
symmetry. Unlike some other models with branes, a preliminary study (not
including the effect of scalar perturbations) shows that the spectrum of
massive Kaluza-Klein states is quantized and free of tachyonic modes.
