I have a custom domain mydomain.net with a CNAME pointing to my Azure Cloud Service at myapp.cloudapp.net. When I enter my custom domain in Firefox it takes me to the azure cloud service and I correctly see my custom domain in the brower location bar. However, when I enter the same custom domain in Internet Explorer (v9) the cloud service loads but my custom domain isn't displayed and I see the myapp.cloudapp.net address instead.
Do I need to do anything else on my custom domain or app? I followed the link here and is says the forwarding should be invisible: http://www.windowsazure.com/en-us/develop/net/common-tasks/custom-dns/#access-app
Do you use CNAME or a domain forwarding?
The article you refer is about CNAME and not domain forwarding (which is different thing).
What you face is most probably domain caching issue for IE. Try deleting all IE caches, wait another couple of hours and try again. There are no issues with CNAME and A DNS Records with Azure.