After much empirical testing, I managed to produce my own solution. BTW, I failed to make the code generic in the original post, it used my own systems paths in the code. This is a more generic version which should work on systems like mine (WinXP SP2, AHK ver. 1.0.47.05)
Code:
UrlDownloadToFile, http://checkip.dyndns.org/, %A_WorkingDir%\ip.txt ; Downloads index.html from the specified URL and saves it as ip.txt in the computers local working directory (eg. C:\TEMP)
ClipSaved := ClipboardAll ; Save the entire clipboard to a variable of your choice.
FileRead, Clipboard, %A_WorkingDir%\ip.txt ; Reads ip.txt into the clipboard.
Clipboard := RegExReplace(Clipboard, "i)[a-z\<\>\:\/]*") ; Strips the letters and HTML tag characters and URL characters leaving only the numeric IP address.
Clipboard = %Clipboard% ; Sets the clipboard as a variable.
FileDelete, %A_WorkingDir%\ip.txt ; Deletes ip.txt (all <1kB of it! Gotta clean up after yourself...)
MsgBox Current IP Address is %Clipboard% ; Opens a message box that tells you what your current IP address is.
Clipboard := ClipSaved ; Restore the original clipboard. Note the use of Clipboard (not ClipboardAll).
I figured most of this out on my own and some "borrowing" from multiple sources, but it would have been much nicer for someone else to have chimed in. I am simply too impatient to wait that long.
I guess the RTFM that I see in some peoples signatures applies, but geez man, it took forever to solve it and even then it was a slow and painful

process.
I updated the final version so that it
should work on any WinXP machine (I hope). Someone let me know if it works on Vista, please.
Thanks.
Jim