Just a heads up that it's not working as expected in Chrome. I loaded the page and got an IP that wasn't mine, then refreshed the page and got a different IP that still wasn't mine. Upon further investigation, both incorrect IPs appear to be owned by CloudFlare.
Same results in Safari. Oddly enough, works as expected in Firefox.
Yeah I was initially testing it on Cloudflare but took it off because I didn't want to fiddle with the server. DNS might be cached for some reason. I switched over to my private nameservers for the time being.
Edit: Are you using 1.1.1.1 by chance? If so turn it off.
Personally I prefer ifconfig.me which takes this idea one step further.
Especially: Querying it with curl (or another system requesting plain text) really only returns the IP, not some HTML page. (Opening it in the browser also shows some useful information about the request, but http://ifconfig.me/ip always shows just your IP address)
I can make a curl version of it at some point. For now, though, the point is that its JUST your IP. Nothing else. It's made for sending grandma's to.
EDIT: Now, best of both worlds. Reload the page and either highlight the page text or look in the source code for how to use it with curl and bash. I'm sure you can figure out your favorite parser for csv.
Also http://v4.ident.me and http://v6.ident.me if you want to check a specific IP version. Unfortunately, leaving off the version prefix (/subdomain) doesn't work.
I like Amazon's page because it doesn't return any HTML, just a simple plaintext response which makes it simple to integrate into a script or bash one-liner. Also, Amazon is large enough that I'm confident they're not going to put in a captcha or rate limit.