  Consider a one-parameter family of smooth, irreducible, projective curves of
genus $g\ge 2$ defined over a number field. Each fiber contains at most
finitely many rational points by the Mordell Conjecture, a theorem of Faltings.
We show that the number of rational points is bounded only in terms of the
family and the Mordell--Weil rank of the fiber's Jacobian. Our proof uses
Vojta's approach to the Mordell Conjecture furnished with a height inequality
due to the second- and third-named authors. In addition we obtain uniform
bounds for the number of torsion in the Jacobian that lie each fiber of the
family.
