Hosting a Tumblr blog on Amazon Web Services as subdomain

How do I use a custom (sub)domain (or secondary page) on AWS to 'host' a Tumblr blog?

I'm using Amazon S3, CloudFront, and Route 53 to host a static website. The static news/blog section I've made lives under mysite.com/news. So far, so good.

Now I've set up a blog/news page on Tumblr (so mysite.tumblr.com) that I want to have replace mysite.com/news. I understand that this might need to change to a two-level structure (news.mysite.com), and that's fine.

So far I've tried following these instructions from Tumblr, but with no luck. I've added a 'Hosted Zone' on Route 53 for news.mysite.com, but failed to point the C-NAME record to domains.tumblr.com.

Basically I'm using Tumblr as a backend for the news page only. Users shouldn't be able to notice any visual difference from any other page. I've already got my HTML & CSS markup perfected.

Any help is much appreciated. Thank you.