  This is a proof of an asymptotic formula which describes exponentially small
splitting of separatrices in a generic analytic family of area-preserving maps
near a Hamiltonian saddle-centre bifurcation. As a particular case and in
combination with an earlier work on a Stokes constant for the H\'enon map
(Gelfreich, Sauzin (2001)), it implies exponentially small transversality of
separatrices in the area-preserving H\'enon family when the multiplicator of
the fixed point is close to one.
