Concurrent H1b

I am currently working for Company A full time with H1b sponsorship. Another company, Company B, gives me a offer which is close to my home and I want to accept it. But Company A wants to continue to hire me for another month or two to complete the project I am working on, and Company A agrees that I could work during weekends. My question is, can I accept payment from Company A after Company B file full-time H1b for me? What does Company A need to do in order to continuously hire me as full-time? and what does Company A need to do in order to hire me as part-time? Thank you so much.