Update 18-sep-2012: Windows Azure Web Sites tiers:
- Free: Allows you to test the Web Site Features
- Shared: Cheap hosting (with 5 GB/month free) that supports both CNAMEs, A-Records and naked domains.
- Reserved: Same as shared, but you get a dedicated VM
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
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