  We say that a contact manifold is Milnor fillable if it is contactomorphic to
the contact boundary of an isolated complex-analytic singularity (X,x).
Generalizing results of Milnor and Giroux, we associate to each holomorphic
function f defined on X, with isolated singularity at x, an open book which
supports the contact structure. Moreover, we prove that any 3-dimensional
oriented manifold admits at most one Milnor fillable contact structure up to
contactomorphism.
  * * * * * * * *
  In the first version of the paper, we showed that the open book associated to
f carries the contact structure only up to an isotopy. Here we drop this
restriction. Following a suggestion of Janos Kollar, we also give a simplified
proof of the algebro-geometrical theorem 4.1, central for the uniqueness
result.
