  We present a brane-world scenario in which two regions of $AdS_5$ space-time
are glued together along a 3-brane with constant positive curvature such that
{\em all} spatial dimensions form a compact manifold of topology $S^4$. It
turns out that the induced geometry on the brane is given by Einstein's static
universe. It is possible to achieve an anisotropy of the manifold which allows
for a huge hierarchy between the size of the extra dimension $R$ and the size
of the observable universe $R_U$ at present. This anisotropy is also at the
origin of a very peculiar property of our model: the physical distance between
{\em any two points} on the brane is of the order of the size of the extra
dimension $R$ regardless of their distance measured with the use of the induced
metric on the brane. In an intermediate distance regime $R \ll r \ll R_U$
gravity on the brane is shown to be effectively 4-dimensional, with
corresponding large distance corrections, in complete analogy with the
Randall-Sundrum II model. For very large distances $r \sim R_U$ we recover
gravity in Einstein's static universe. However, in contrast to the
Randall-Sundrum II model the difference in topology has the advantage of giving
rise to a geodesically complete space.
