The story is false.
There used to be a poster on FR who was very knowledgeable on military matters. He posted his opus and said he was never coming back to FR. I was just about to do the same thing but FR got rid of the ones who were banning conservative posters and I am glad I stayed.
I wish he would come back but he said he never would and was true to his word.
This guy, who a lot of people here remember well, mentioned that he actually met Zaitsev in Russia. He said as a matter of fact, that Zaitsev himself never claimed the story was true. It was finally proven when after the war, German records showed there was no such man as Major Koenig from the German Sniper School.
The story has been published as fact in many articles and probably historical documents, but it was not true. I can’t blame anyone who believes it as I was one of them.
There is a small snippet you might find interesting: http://www.youtube.com/watch?v=g3QsBUoX2aQ