We have an Azure Website hosting a static site (just some HTML, CSS, Javascript), which then communicates with our Azure Mobile Services by AJAX calls.
We would like to
This is an old question that's been previously answered, but if you're looking for the easiest possible solution that requires no code or config changes, you can use Azure Websites Authentication / Authorization (disclaimer, I helped build this feature). Here are some links:
Blog Post: http://azure.microsoft.com/blog/2014/11/13/azure-websites-authentication-authorization/
Demo Video: http://azure.microsoft.com/en-us/documentation/videos/azure-websites-easy-authentication-and-authorization-with-chris-gillum/
EDIT: And here is a more recent blog post describing how you can use social providers (Twitter, Facebook, Google, Microsoft Accounts) in addition to the default Azure AD: https://azure.microsoft.com/en-us/blog/announcing-app-service-authentication-authorization/