  A classification of supersymmetric solutions of five dimensional ungauged
supergravity coupled to arbitrary many abelian vector multiplets is used to
prove a uniqueness theorem for asymptotically flat supersymmetric black holes
with regular horizons. It is shown that the near-horizon geometries of
solutions for which the scalars and gauge field strengths are sufficiently
regular on the horizon are flat space, AdS_3 x S^2, or the near-horizon BMPV
solution. Furthermore, the only black hole which has the near-horizon BMPV
geometry for its near-horizon geometry is the solution found by Chamseddine and
Sabra.
