In the current technical preview, Azure lets you create 10 ASP.NET websites. They are given domains such as http://yourappname.azurewebsites.net. Is there currently any way to p
For what it's worth, it's now (finally) available for about $10/month, in shared and Reserved modes). I've tested it and it's working perfectly, after the little DNS dance for the naked domain. This was the missing piece to Azure for some small sites I run.
The Gu has all the info: http://weblogs.asp.net/scottgu/archive/2012/09/17/announcing-great-improvements-to-windows-azure-web-sites.aspx
Update 18-sep-2012: Windows Azure Web Sites tiers:
Read all about it in Scott Guthrie's post: Announcing: Great Improvements to Windows Azure Web Sites
Original Answer:
At the moment this is only possible for reserved instances (using a CNAME), this means it's not possible for the free sites you can create (which are shared):
You can only used custom domains in reserved mode. You can’t set a CNAME like you can in Web Roles because multiple sites share the same IP address. Switch to reserved mode and set your CNAME in the configuration and it should work.
http://blog.ntotten.com/2012/06/07/10-things-about-windows-azure-web-sites/
Microsoft confirmed that they will add support shared sites in the future:
https://twitter.com/scottgu/status/210972290719031298
You will soon be able to use Custom Domain without having to upgrade your plan to RESERVED. We are introducing a new Shared mode. The new Shared mode will have some cost, but it is a lot less than reserved mode, and it will support Custom Domain. Please check back at the http://www.windowsazure.com/en-us/ soon for more info.
As mentioned by Sandrino, this is possible only for reserved instances.
Not sure about Scott Guthrie answer. Check this thread.
It is stated that at this time there is no plans to support custom domains for shared instances.
It also contains a request to comment on Scott's answer.
Update:
It might be available for a fee (same thread)
Fast forward two years, it is still not possible to have CNAME directed to Azure free website. The reasons are commercial and not technical. MS says that it is not possible because multiple sites share a same IP. But don't know why I have this dedicated server at GoDaddy that has one IP but hosts 7 different domains :-) So it can't be a technical reason. Of course, if they start allowing it, nobody would want to subscribe to paid versions.