can i legally stay?

I am holding a J1 visa right now and it will expire at the end of this month. After this I still have 30-day grace period. And the company already filed H-1B application for premium processing on my behalf. If it is approved, the earliest time I can start to work with H1B is Oct. 1. So there will be a 2-month (Aug.1 - Sept. 30) gap. I don't know if I can legally stay in US during this period. Anybody has this kind of knowledge, please give me some info. I really appreciate it.