I don't know if this is exposed via .NET, but if it is you're looking for  
"Universal Plug and Play" (aka "UPnP").  Probably in the WMI classes.

Keep in mind that just knowing your Internet IP address is unlikely to be  
sufficient.  If you really need to know that address (and there's a good  
chance you don't), you probably also need to configure the router  
identified via UPnP to forward traffic sent to that address on a specific  
port to your own local host computer and application.

Other than "how" to get at the UPnP information, this is really more of a  
networking question than a .NET question and you'll probably get better,  
more detailed answers in a newsgroup dedicated to networking questions.

Pete

Your public IP is often assigned by your ISP--to which you likely have no
access.

There's various web-sites out there that will echo back your IP, for example
http://www.whatismyip.com/