What happens if you move your king on stalemate? Let's say the following happens during a blitz match:


White makes a move, resulting in stalemate
Black doesn't notice the stalemate and moves the king into an illegal position.


Usually illegal moves result in instant loss according to FIDE rules, but a stalemate results in instant draw. So, what's the outcome of the game? Draw or win for white?

This didn't happen in one of my matches, but 
I've had some losses in the past because of illegal moves and I'm wondering what would happen in this case.
Just like checkmate, stalemate immediately ends the game. From the FIDE Laws of Chess, 5.2a:


  The game is drawn when the player to move has no legal move and his king is not in check. The game is said to end in ‘stalemate’. This immediately ends the game,
  provided that the move producing the stalemate position was legal.


This is not overridden by any rules in Appendix B, covering Blitz, so the situation you describe is a draw.