  Using mixed numerical and analytical methods we give evidence that the 6+1
dimensional Taub-NUT soliton is asymptotically nonlinearly stable against small
perturbations preserving biaxial Bianchi IX symmetry. We also show that for
sufficiently strong perturbations the soliton collapses to a warped black hole.
Since this black hole solution is not known in closed form, for completeness of
the exposition we prove its existence and determine its properties. In
particular, the mass of the black hole is computed.
