Why am I getting 2 IP addresses from the cable modem?

I recently renewed my IP address for my cable modem (by booting into Windows and doing the ipconfig thing). It did this successfully and assigned me a new static IP.
However back in OS X I am now getting two IP addresses depending on whether I connect by ethernet or Airport. When I connect my MacBook straight to the cable modem by ethernet it gives me the new IP address. However when I connect the Airport Extreme Base Station to the cable modem and do all the relevant restarting, it insists I still have the old IP address. This can't be right. Is it an errant preference hanging about in the Base Station? I have tried restoring it to default and setting up from scratch but it persists.
I know the difference between public and distributed addresses... the two I'm talking about are the public kind (e.g. 82.114.223.43 etc)

What do you mean by "old" and "new" IP addresses? What TYPE of IP addresses are you talking about? Specific numbers for example?
In general, the typical cable modem set up should work as follows: your ISP will assign an IP address to your cable modem. This IP address can be permanently fixed (static) or occasionally change (dynamic) - whether you get a static or dynamic IP address depends on your ISP. This IP address is considered your "public" IP address.
What you plug in directly after the cable modem will take on this IP address, since the cable modem merely passes on the IP address info (and serves no routing functions). So, if you plug in your MacBook directly to the modem, your MacBook will take on the IP address that your cable modem passes through, as assigned by your ISP; only your MacBook will have internet access.
On the other hand, if you plug in to the modem a router, such as Airport Extreme Base Station, the router will be assigned the "public" IP address received from the cable modem on its WAN (wide area network) side, i.e. the "side" that interfaces with the public Internet. When your router is set up to SHARE this one public IP address with multiple computers/clients, it has to create a private sub-network, otherwise known as the LAN (local area network). Whether you plug computers in via ethernet to the router, or connect wirelessly to the router (or even both!), your router will create IP addresses for each of its LAN clients so that it can identify everyone. As long as the client has a different hardware MAC address, the router will assign it an unique IP address. (That's why you had 2 IP addresses for your MacBook - one for the ethernet connection, and the other for the Airport connection - each network interface having a different MAC address, and thus receiving a different IP address from the router.)
Your router will always maintain its WAN IP address, as long as your ISP gives you that IP address (via your cable modem). However your MacBook, Windows and other clients may be assigned different LAN IP addresses depending on how you configured your router's DHCP server settings. The LAN clients in the context of your subnet would NOT be assigned the router's public WAN IP address when the router is set up to share that connection.
If you understand the whole WAN v. LAN IP addressing thing, but you're not getting the results you expect, try powering off EVERY single piece of equipment on your network. Wait several minutes so that everything is cleared from the equipments' memory. Then power on and wait until the equipment is in ready state, in this sequence: 1-cable modem..., 2-main router (AEBS)..., 3-remote bases/bridges (if any)..., 4-computers and other client devices.

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