Hi
I've also been looking for a flat in Zone 1 and not had any luck so far. I have found Craigslist and Gumtree often full of gorgeous flats at incredibly low prices that are just too good to be true though, and I also read this article this weekend.
http://www.telegraph.co.uk/travel/travelnews/8639513/Tourists-duped-by-holiday-rental-scams.html
So, good luck, but be careful too!
R xx