  We study asymptotically AdS generalizations of Melvin spacetimes, describing
gravitationally bound tubes of magnetic flux. We find that narrow fluxtubes,
carrying strong magnetic fields but little total flux, are approximately
unchanged from the $\Lambda=0$ case at scales smaller than the AdS scale.
However, fluxtubes with weak fields, which for $\Lambda=0$ can grow arbitrarily
large in radius and carry unbounded magnetic flux, are limited in radius by the
AdS scale and like the narrow fluxtubes carry only small total flux. As a
consequence, there is a maximum magnetic flux $\Phi_{max} =
2\pi/\sqrt{-\Lambda}$ that can be carried by static fluxtubes in AdS. For flux
$\Phi_{tot}<\Phi_{max}$ there are two branches of solutions, with one branch
always narrower in radius than the other. We compute the ADM mass and tensions
for AdS-Melvin fluxtube, finding that the wider radius branch of solutions
always has lower mass. In the limit of vanishing flux, this branch reduces to
the AdS soliton.
