  We show that AdS black hole solutions admit an analytical continuation to
become magnetic flux-branes. Although a BPS AdS black hole generally has a
naked singularity, the BPS flux-brane can be regular everywhere with an
appropriate choice of U(1)-charges. This flux-brane interpolates from AdS_{D-2}
\times H^2 at small distance to an asymptotic AdS_D-type metric with an
AdS_{D-2}\times S^1 boundary. We also obtain a smooth cosmological solution of
de Sitter Einstein-Maxwell gravity which flows from dS_2\times S^{D-2} in the
infinite past to a dS_D-type metric, with an S^{D-2}\times S^1 boundary, in the
infinite future.
