See also the related
The use of embedding a curve into its Jacobian
If you don't have access to Timo's reference there is also the free resource
J.Stix On cuspidal sections of algebraic fundamental groups http://arxiv.org/abs/0809.0017 Appendix B
Beware that this appendix was suppressed from the published version (eponymous article on Jakob's webpage).
And yes, the argument is due to Grothendieck. You can find the original letter (translated from German into English) here :
http://www.math.jussieu.fr/~leila/grothendieckcircle/GtoF.pdf
p. 282/4
"The proof follows rather easily from the Mordell-Weil theorem stating that the group $A(K)$ is a finitely generated $\mathbb Z$-module, where $A$ is the “jacobienne généralisée” of $Y$ , corresponding to the “universal” embedding of $Y$ into a torsor under a quasi-abelian variety."