How can I quit my job at such a short notice?

I accepted a job offer and they want me to start this Wednesday and I want to finish the rest of this week at my current job. My current job is at a supermarket market I been at the company for almost 2 years now then I was transferred to one of their other stores when I relocated 2 months ago so I been at the new store for 2 months. I know the right thing to do is to give a 2 weeks notice but the new job didn't give me an official start date.The current job gave me crappy hours between 12-20 per week and hubby and I struggled to pay the bills so I see no point being there any longer.