  We present black hole uniqueness theorems for the C-metric and Ernst
solution. The proof follows a similar strategy as that used to prove the
uniqueness of the Kerr-Newman solution, however the presence of an acceleration
horizon provides some critical differences. We also show how to derive the
Bunting/Mazur result (on the positivity of a suitable divergence required in
the proof) using new methods. We briefly explain the importance of the
uniqueness of the Ernst solution in relation to the proposed black hole
monopole pair creation mediated by the related instanton.
