I noticed tonight that my Cisco 837 was using hard-coded DNS servers which were no longer valid, hence breaking my internets. In order to keep them updated automatically, I added the following to the appropriate dialer configuration (interface Dialer0 on an 8xx):

This causes the router to request DNS server addresses when the PPP session is negotiated (when it “dials up” to ADSL).

After reconnecting the PPP session (either reload or shutdown the interface and then bring it back) it is possible to check that DNS servers have been obtained using sh host:

If the router is running a DNS server it is possible to direct DHCP clients at the router IP instead, forcing them to use these same settings.