See also the related 

https://mathoverflow.net/questions/92147/the-use-of-embedding-a-curve-into-its-jacobian/92157#92157

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."