Chip,

First of all, it sounds like your router may have a serious problem. It should not reassign the same IP address.

Second, are you absolutely sure that you are connecting to your router and not another router in the neighborhood? Yea, it seems like a stupid question, but it happens more often than you would expect. Much more often than a router reusing an IP address!

Third, what make and model of router is this? That will be a big factor in what settings you have to deal with this. For example I like Dlink because they let me lock a MAC address to an IP address and still give the IP addresses out by DHCP. So I can be sure that once I have the table set up a particular MAC will always get the same IP address. Handy for when you run servers that need ports forwarded to an IP address, and Dlink is the only brand that I have found that I can do this with and still use DHCP on the same computer (which, as a Knoppix user, I want to do).

Forth, in addition to the router make and model it might also help if you were very explicit about the addresses being handed out. As long as these are private addresses they are completely unreachable from the Internet, There is no security reason not to state the addresses explicitly, and since something is clearly going wrong here it might help us if we had all of the facts.