I have worked past the end of my employment contract - can I quit?

My one-year employment contract finished at the end of March, but I have continued to work there without signing another contract. I have just received a job offer from another company. The (now expired) contract stipulates that I must give three months notice if I want to quit. However, I had assumed that as long as I had not signed a new one, then I could quit at any time. If I were to tell my current employer that I have no intention to work more than one month's notice, would they be entitled to take legal action against me?