"A proof of projective determinacy" by Martin and Steel, Journal of the American Mathematical Society, 2(1):71–125, 1989, although it may not look quite as what we are used to think of these things now. 
And probably you want to read first Martin's proof of determinacy of $\Pi^1_1$ from sharps, since the key ideas are there (this should be in Jech's or Kanamori's book). 

A modern, very quick and nice exposition of this and the key related ideas in proving determinacy at the projective level is in Itay Neeman's paper "Determinacy in $L({\mathbb R})$", in the Handbook of Set Theory. You can currently download the paper from Itay's [page][1].

A thorough and superb presentation of weak homogeneity and much more, including the result you are asking for, is in "The Derived Model Theorem", an unpublished note by John Steel; the latest version is dated May 29, 2008, and can be downloaded from his [page][2]. 


  [1]: http://www.math.ucla.edu/~ineeman/
  [2]: http://math.berkeley.edu/~steel/papers/Publications.html