  We present a possible explanation for the smallness of the observed
cosmological constant using a variant of the Randall-Sundrum(RS)-
Goldberger-Wise paradigm for a warped extra dimension. In contrast to RS, we
imagine that we are living on the positive tension Planck brane, or on a
zero-tension TeV brane. In our model there are two solutions for the scalar
field in the bulk and the corresponding brane separations, one of which is
tuned to have zero cosmological constant. We show that in the other solution,
which is a false vacuum state, the 4-D cosmological constant can be naturally
small, due to exponential suppression by the warp factor. The radion is in the
milli-eV mass range, and if we live on a TeV brane its couplings are large
enough that it can measurably alter the gravitational force at submillimeter
distances. In this case the Kaluza-Klein excitations of the graviton can also
contribute to submillimeter deviations from Newtonian gravity, and we have in
addition the collider phenomenology of the usual TeV-scale radion.
