How to put a router with DHCP between my computers and the modem while keeping the 192.168.0.x addresses?

My internet provider has occasional issues and the modem that is a DHCP server for my computers leaves me without a local network until their issue is resolved. So I figured if I put a router (with a DHCP server and disable the DHCP server on the modem) between the computers and the modem I can keep my LAN on while the modem is off. I don't want to change the IP addresses of the computers that are now 192.168.0.x so how do I do that correctly?

Question by B Viana on Super User, licensed CC BY-SA. We show a short excerpt and link to the original thread as credit.