  Let $A$ be a non-zero abelian variety over a field $F$ that is not algebraic
over a finite field. We prove that the rational rank of the abelian group
$A(F)$ is infinite when $F$ is large in the sense of Pop (also called ample).
The main ingredient is a deduction of the 1-dimensional case of the relative
Mordell-Lang conjecture from a result of R\"ossler.
