If your H-1B is subject to the cap, you can start work until October 1. You must not stay in the U.S. unlawfully while you are waiting to start work.
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Thank you for your inquiry. I will attempt to answer the inquiry, but some of the answer depends on the employer who is submitting the petition for you and unfortunately, you have not provided enough information in the inquiry to assess that part of the question.
So the deal is that there are 65,000 H-1B visas made available every government fiscal year. In addition to the 65,000 visas above, there are 20,000 visas which are made available to certain individual who have attained a masters degree or higher from certain US colleges and universities. These numberical limitations are commonly referred to as the
H-1B cap and the H-1B cap generally confuses the heck out of everyone involved in the discussion.
Now, the visas which are made available are government by the US government's fiscal year ("FY"). The FY starts on October 1 and ends the following September 30, So right now, we are in the middle of FY 2010. The new quota of H-1B visas will be available for employment start dates on or after October 1, 2010 - the start of FY 2011.
H-1B's can be applied for 6 months in advance of the employment start date. So today is April 2, 2011, which means that if an employer wants to obtain an H-1B visa subject to the FY 2011 cap, the window is now open for those visas. BUT, even if an application for this visa is approved, the employment start date CANNOT begin until October 1, 2011.
So, if you are applying for a FY 2011 visa and you are currently here on a B-1, you have to maintain lawful non immigrant status until October 1, 2011 if you want to remain in the US and be permitted to work on an approved H-1B come October 1, 2011. My guess is that you will not be here for another 6 months as a visitor. With that said, I think that the answer to your question is that you will likely be required to return back to your home country to obtain an approved H-1B visa.
Wow, that is complicated and makes absolutely little sense, but this is the deal. Some H-1B applications, however, are not subject to a cap. Certain employers can offer employment through an H-1B and not be subject to the cap. This is why I explained above that you may be able to remain in the US and obtain an H-1B depending on the employer. Note, however, that these "cap exempt" visas ar sometimes difficult to obtain and whether your employer may qualify for one is a question which can be answered by a competent immigration attorney.
I am not sure if you have hired an attorney to handle the H-1B application for you. But to the extent that you have, that attorney should hopefully have the understanding and experience to answer your question. If you are not receiving the responses that you are seeking, then find someone who can answer your questions. There are many restrictions and rules that apply to the H-1B visa program and having a qualified legal professional on your side is indeed a blessing.
Hope I could answer some of your questions, if you have more please consult with someone in the know. Good luck.