  Investigations of classical signature change have generally envisaged
applications to cosmological models, usually a
Friedmann-Lemaitre-Robertson-Walker model. The purpose has been to avoid the
inevitable singularity of models with purely Lorentzian signature, replacing
the neighbourhood of the big bang with an initial, singularity free region of
Euclidean signture, and a signature change. We here show that signature change
can also avoid the singularity of gravitational collapse. We investigate the
process of re-birth of Schwarzschild type black holes, modelling it as a double
signature change, joining two universes of Lorentzian signature through a
Euclidean region which provides a `bounce'. We show that this process is viable
both with and without matter present, but realistic models -- which have the
signature change surfaces hidden inside the horizons -- require non-zero
density. In fact the most realistic models are those that start as a finite
cloud of collapsing matter, surrounded by vacuum. We consider how geodesics may
be matched across a signature change surface, and conclude that the particle
`masses' must jump in value. This scenario may be relevant to Smolin's recent
proposal that a form of natural selection operates on the level of universes,
which favours the type of universe we live in.
