Let K be a number field, S a finite set of places of K, \(R_ S\) the ring of S-integers, and \(A| K\) an abelian variety. A conjecture of \textit{S. Lang} [''Fundamentals of diophantine geometry'' (1983; Zbl 0528.14013); p. 219] asserts, for every affine subset U of A, the finiteness of the set of points of U with coordinates in \(R_ S\). In the paper under consideration, a first step towards this conjecture is proved. (*) Assume \(A=J(C)\) is the Jacobian of a curve C of genus \(\geq 2\) over the number field k. There exists a positive, irreducible divisor \(D\in Div(A)\) such that for every finite extension K:k, every finite set S of places of K, every \(f\in K(A)\) with pole divisor \((f)_{\infty}\geq D\), the set \(\{x\in A(K)| f(x)\) is defined and lies in \(R_ S\}\) is finite. Corresponding finiteness results can be given for arbitrary abelian varieties A, using a finite map of A into a Jacobian. The proof of (*) is straightforward, the ingredients being (a) a criterion of good reduction of curves which the author assigns to Szpiro and Ogus, and (b) (the main point), Faltings' proof of the Shafarevich conjecture.