Company released me immediately after resigning and not paying for not

I work for a private firm which has a policy of putting 1 month notice period if any employee resigns, or if the company decides to terminate the employee, company will have to give 1 month notice to employee. Failing to which, either party will have to pay 1 month salary. In my case, I recently joined this company 2 months back, but I felt that this is not the kind of work I would like to do, so started looking for change. I got a verbal offer and decided to give 1 month notice to my current employer, simultaneously the company from where I had got a verbal offer gave me a joining date of 1 month later, thinking that my notice period will be served in this 1 month. Now when I resigned, I told them that I'll serve 1 month notice period. But the current company accepted my resignation from immediate effect and told me that due to business shortage, it would be my last working day. They are not ready to pay for this notice period days.