In fairness, you do have contract with your employer and yes in theory you do have to give 2 weeks notice.
If you have a lot of annual leave allowance left, SOME employers may take that into acount and deduct that from your notice period.
In fairness, depending on what you do for a living, during your notice period you may be required to do a handover to the new person taking your place therefore leaving so suddenly may not allow you current employer to back fill your current position.
If you have good relationship with your line manager you may be in a position to negotiate. You can only try.
The only other option you have is to explain this to your new employer. If they want you bad enough I am sure they can wait 2 weeks, and as an employer myself , I allow people about to join my organisation, time to serve their notice, hoping that should they want to leave my team, they'd give me back the same level of commitment/respect.
Good luck!