  We consider a blown-up 3-brane, with the resulting geometry R^(3,1) \times
S^(N-1), in an infinite-volume bulk with N > 2 extra dimensions. The action on
the brane includes both an Einstein term and a cosmological constant. Similar
setups have been proposed both to reproduce 4-d gravity on the brane, and to
solve the cosmological constant problem. Here we obtain a singularity-free
solution to Einstein's equations everywhere in the bulk and on the brane, which
allows us to address these question explicitely. One finds, however, that the
proper volume of S^(N-1) and the cosmological constant on the brane have to be
fine-tuned relatively to each other, thus the cosmological constant problem is
not solved. Moreover the scalar propagator on the brane behaves 4-dimensionally
over a phenomenologically acceptable range only if the warp factor on the brane
is huge, which aggravates the Weak Scale - Planck Scale hierarchy problem.
