Change of OPT status to H1B. OPT expires in a week

My OPT expires May 15th 2013, I have a job offer. How can i ensure that my company files the paperwork in time so as not to lapse on my visa. Would I have to leave the US and could I begin working with them immediately if they apply before my oPT expiration date?

I am assuming that you did not hold H1B previously.

Has your current company filed for H1 in this cycle (2013)?

If yes, then has it been approved?

If Yes, then you will need to apply for CAP-GAP which allows you to stay in US but not work from - May 15th to Oct 1st ( If this is your first time H1 then you will need to wait anyways till 1st of OCT irrespective of you taking up new job or not)

If H1B was not applied in this cycle - nothing can be done now as H1B quota is over for this year. You will need to wait till next year. And most probably go back to india.