I think it is a viable option. However, to get your own domain name you must change the website from free to shared or reserved mode. Heres description and link how to do this!
"When you create a web site, Windows Azure provides a friendly subdomain on the azurewebsites.net domain so your users can access your web site using a URL like http://.azurewebsites.net. However, if you configure your web sites for shared or reserved mode, you can map your web site to your own domain name."
http://www.windowsazure.com/en-us/develop/net/common-tasks/custom-dns-web-site/