Dear @Gentle james,
thank you for sharing your question with the Wefugees Community.
First of all, I would like to congratulate you for your new work. Coming to your question, you can talk to your old employer and ask them to make with you an Aufhebungsvertrag (agreement to terminate your contract). If your old employer agrees with you about it (and I think they will; because, as you said there is no work there and the company is close), you or your employer does not need to stick to the one month 's notice. In this case, you can end your employment contract with your old employer immediately.
If you have two jobs at the same time and same hours of work, you might face problems because of your taxes and so on. However, if you still want to choose this option, talk to your old and new employer about it.
I hope my answer was useful for you and I'll be very pleased if you contact me again with further questions. I'm a member of the MBEON-staff and I‘ll be glad to guide you over a longer period of time. Mbeon is an online consultancy for migrants via the mbeon-app, which you can install on your smartphone for free. Download it via App Store or Google Play Store. You'll find me In the mbeon-app under my name „München, Herr Noori“ and get confidential one-to-one advice in a safe virtual room where nobody can wiretap messages. Further information: www.mbeon.de.