  We discuss the possibility of localizing Einstein gravity on a three-brane
embedded in a five-dimensional Minkowski space by giving a space-dependent mass
to the graviton. We show that, with an overall fine tuning of the mass profile,
it is possible to localize a massless graviton with a finite mass gap from the
massive Kaluza-Klein modes. The theory does not contain ghosts and no scalar
massless mode is sourced by brane matter. A massless vector mode is strongly
coupled to the flux of energy-momentum into the extra dimension, so that it is
not sourced as long as only brane-localized matter is considered. In the case
where the mass profile has a delta-like shape the four-dimensional Planck mass
$M_4$ is related to the five-dimensional one, $M_5$, by $M_4^{2}=M_5^3/m_0$,
where $m_0$ is the mass of the first Kaluza-Klein mode of the graviton.
