Tearch 8, Pijiang, and all experts, emergency question regarding

My current H1B (from a university) will be expired on August 1st. Now I got a job offer from a local industrial company, which is willing to sponsor a new H1B. The immigration lawyer of the company told me that, as long as the company file the application for me, I should be able to work for the company immediately and legally.

However, that sounds confusing to me because the H1B of industry has a cap of 65,000/year. If the quota of year 2011 is totally used out, then I can only start to work on October 1, 2011 legally, which is the starting date of H1B of 2012. Am I right? In addition, if the lawyer is right, then after I submit the H1B application, what is the my exact visa status between August 1 and October 1? Is it stilled counted as H1?

Thank you for your help!