Where can I read a corrected up to date version of Heegner's solution of the class 1 problem of Gauss?
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The wikipedia entry for Kurt Heegner contains a reference to an article by Harold Stark where he adresses the gap on the original proof and presents an outline of Heegner's argument.
Heegner's original article (cited in the wikipedia entry) seems to be available in SpringerLink if you have access to it.
Another nice reference which you would like to consult is David Cox's excellent book Primes of the form $x^2 + ny^2$. In section 12 of that book you can find a proof under the entry E. Imaginary Quadratic Fields of Class Number 1. I hope this helps.
There is an article of Birch from 1969, titled Weber's class invariants, in which he proves, using basic class field theory and CM theory, various facts originally due to Weber, but whose proof was in doubt. These are facts that Heegner cited in his proof, and Birch observes that one reason Heegner's proof was doubted. Given this, it seems that Birch's article will be a useful companion when reading Heegner's argument.