AirPort Extreme: Keep IP address from changing
Your External IP Address (or, as you call it, the IP address reported by whatismyip.com) is not under your control and you can't simply set it to static. Your ISP gives you a new one when you reconnect, although there are chances that you might get the same, you should never count on it.

In order to solve this problem you have two choices:


Some Internet Service Providers (ISPs) offer you a "Static" IP service for a monthly fee. With the lack of IPv4 in the world, this service is unlikely to exist forever to individuals and it's usually reserved for companies and services that really need a static IP. Not all the ISPs offer this, you can always ask.
Use a Dynamic DNS provider. This will mean that although your IP address will eventually change when you disconnect (or your provider wants), you will always be able to get "back" to your Airport's IP by using a "name" rather than an IP. Any computer in your network will have a small piece of soft (some routers can do this automatically, I'm not sure about Airport Extreme, don't think so), that will periodically check your IP and if it has changed, it will inform the Dynamic DNS service that you use so it can update the value.


Some Dynamic DNS Providers (Mostly a free service) are (in no particular order): 


DynDNS
No-IP (I use this)


Think of this as if instead of having a fixed phone number (static ip), a person could call you by "name" and the phone will check what is your current number (wouldn't that be great? ;) Nobody knows your number, only your name, because your number changes often (your ip!).

You haven't mentioned why you need a Static IP in the first place, but if you can't get it via your ISP, your only hope is to use Dynamic DNS. For what is worth, DDNS works really fine and it's very "fire and forget". Once it works, you no longer have to worry about IPs or anything, you just connect to: XXXXX.no-ip.org (for example)