  Let s be a special point on a Shimura variety, and x a pre-image of s in a
fixed fundamental set of the associated Hermitian symmetric domain. We prove
that the height of x is polynomially bounded with respect to the discriminant
of the centre of the endomorphism ring of the corresponding Z-Hodge structure.
Our bound is the final step needed to complete a proof of the Andre-Oort
conjecture under the conjectural lower bounds for the sizes of Galois orbits of
special points, using a strategy of Pila and Zannier.
