Notice period in a company gives time to employer for searching a replacement of the existing employee, who is about to leave, and an employee to hand over the overall charge like files, documents, information etc. to the department.
Legally you will have to serve one month notice period with the current employer and there will be some clause in your job agreement. It may be serving one month notice period or surrender salary of one month. On failing to follow the rules of the agreement, your current employer is free to legally sue you.
But as you have already faced interview with some other employers and got offer from big company. Now there are two options:
1. You can negotiate with your current employer. You can talk to your employer and can explain them your situation and can also request for early relieving like one week or 2 week. Further, you can ensure them that you will do work honestly and hard and will handed over all the documents and information to the new or existing employee sincerely. Your employee may allow for early relieving as it will save their time and money also.
But I think doing conversation with the current employer might be risky to some extent because, you have not signed agreement with new employer yet. You may be fired by your current employer on hearing about new job offer and you never know about new employer.
2. You can talk to your new employer about the notice period of current job. May be they extend your joining or they may surrender one month salary to your current employer on your behalf.
You should think carefully and talk to your colleague and HR department about it. However, already you have an offer in your hand and if you want to join the new job, you can talk to head of the Department or HR department and also you can give a shorter notice period and leave the job earlier.