There must be a proof in  Cassels' *Lectures on elliptic curves* (Cambridge University Press, Cambridge, 1991).

Se also his [masterly survey][1] *Diophantine equations with special reference to elliptic curves* (J. London Math. Soc. **41** (1966) 193–291) and the [historical essay][2] *Mordell's finite basis theorem revisited* (Math. Proc. Cambridge Philos. Soc. **100** (1986), no. 1, 31–41).

Here is a quote from this last paper :

*Weil's generalization of Mordell's theorem (and subsequent generalizations) was usually referred to as the Mordell-Weil Theorem. Mordell himself strongly
disapproved of this usage and frequently insisted (in public and in private)  that what
he had proved should be called Mordell's Theorem and  that everything else could, for
his  part, be called simply Weil's Theorem.*

**Addendum**.  Another excellent source is Knapp's *Elliptic curves*
(Princeton University Press, Princeton, 1992) which contains a proof of Moredell's theorem (over $\mathbf Q$).


  [1]: http://jlms.oxfordjournals.org/content/s1-41/1/193.full.pdf
  [2]: http://journals.cambridge.org/action/displayFulltext?type=1&fid=2092256&jid=PSP&volumeId=100&issueId=01&aid=2092248&bodyId=&membershipNumber=&societyETOCSession=