Do you have a contract from your current employer, and what does that stipulate? (quite often in the first 6 or 12 months, you are considered 'on probation' and have a much shorter notice period.
Notice periods are usually protection for you from being suddenly out of work (either fired or redundancy). They are also an indication of how much notice a company would expect you to give them before leaving.
However, a company cannot make you stay working for them. It's not in their interest to have a disgruntled employee forced to work for them when they don't want to be there, and you can usually reach a mutual agreement that suits both parties.
Provided you have got a contract for the new job, and all your references have been checked, then you could could either phone in sick, or even just not turn up for work your current company. (or you could turn up and do nothing all day - what are they going to do, fire you?)
I would suggest talking to the company and explaining that you will be leaving on X date as you are leaving to start a new job, and it's in nobodies interest for you to stay there any longer.
I've worked a shorter notice period a few times as certain things haven't always lined up, and it's never been an issue. When I've written my formal resignation letter I've also stipulated my last working day, you can do it with or without mentioning any agreement with your line manager:
As agreed with [X/my line manager] my last working day will be x/x/x
As I am leaving to start a new job elsewhere, my last working day will be x/x/x
Bottom line here is talk to them first, but don't give them any room to manoeuvre. Say what your last day will be, stick to it, and make it clear (politely, every time) that is the last day you will be turning up for work.