  This is the second article of a series or two, proving a generalisation of
the uniqueness theorem of the Schwarzschild solution. The theorem to be shown
classifies all (metrically complete) solutions of the static vacuum Einstein
equations with compact but non-necessarily connected horizon without any
further assumption on the topology or the asymptotic. Specifically, it is shown
that any such solution is either: (i) a Boost, (ii) a Schwarzschild black hole,
or (iii) is of Myers/Korotkin-Nicolai type, that is, it has the same topology
and Kasner asymptotic as the Myers/Korotkin-Nicolai black holes. In this Part
II we show that the only end of a static black hole data set is either
asymptotically flat or asymptotically Kasner. This proves the third and last
step required in the proof of the classification theorem, as was explained in
Part I. The analysis requires a thorough study of static data sets with a free
$S^{1}$-symmetry and a delicate investigation of the geometry of static black
hole ends with sub-cubic volume growth. Many of the conclusions of this article
are hitherto unknown and have their own interest.
