Problem solved. AT&T changed the way DSL works just enough that the 7-year-old modem mo longer works correctly. They sent me a new modem, and the new modem contains a router. There is only one set of addresses for both.
The new modem also has a SETTING for disabling DHCP IP generation. Neither the old modem nor the old router has such a setting in its menu (or it's an acronym I don't recognize as that).
I am going to leave this open for further suggestions, as I still want to learn about this.