Extended stay while H1B application

My L1B validity : Till June 2012.
i-94 date : Till May 2013.

Currently working here on L1 since have a valid I-94 date. My employer is planning to apply for my H1B . Can i stay here after May 2013 if i have my H1B petion applied.

If you are applying H1 for the first time, you cannot work after May 2013 till end of Sep 2013.

But you can stay until you know the result of your H1 petition without working.