If you own one or more domains via the Route 53 registry in AWS, you can easily use it for your WordPress.com website or blog. You will need a paid plan in order to use custom domains.
Using a Paid Plan
If you have a paid plan you can use WordPress.com as your DNS provider so you will only need to edit the Name servers in Route 53.
Step 1: Add Domains to WordPress.com
Start by mapping your domain(s) to the WordPress.com website via Manage > Domains > Add a domain > Use a domain I own. (See WordPress Support for more information)


Step 2: Remove Your Hosted Zone in AWS
If you registered a domain in AWS Route 53 it is likely that a hosted zone was automatically generated when you first registered. Simply navigate to your hosted zone in the AWS Console and delete it.
(Remember to copy the information here just in case you want/need to recreate the hosted zone later)

Step 3: Add the Name Servers to Registered Domains
Navigate in Route 53 to Registered domains and click on your domain.
Click the Add or edit name servers button.

(Remember to copy the current name servers in case you need to revert it back to AWS name servers.)
Delete the current name servers, and replace them with the WordPress.com name servers:
ns1.wordpress.com
ns2.wordpress.com
ns3.wordpress.com

Step 4: Verify Changes have Taken Effect
Once you have done all this, it will take AWS upwards of 48 hours to replicate the changes across the entire environment.
You can check back in the WordPress.com domain management to see if the domain is correctly linked. You should see a green “Active” confirmation if everything is working.

Using a Free Plan
It is not currently possible to use a custom domain to point to your yourgroovydomain.wordpress.com free domain. The website will block any cross domain queries.
You should expect an error like this:

If you have any ideas or suggestions. Let me know in the comments, or contact me via the contact form.