How can I use Cloudfront with a root domain name?

I set up a Cloudfront distribution. I use a non-AWS domain registrar and DNS. I want my distribution to respond to "https://mydomain.com", but there is a problem. Cloudfront provides a domain name and asks you to create a CNAME record in DNS, but you can't create a CNAME record that points to the root domain or "@", like you can with a regular A record.

To get around the problem, I set up "www.mydomain.com" as the CNAME record. If I type "https://www.mydomain.com" into my browser it works, but of course "mydomain.com" without "www" does not work.

The next thing I did was create a permanent redirect in DNS that should redirect mydomain.com to www.mydomain.com. Now I can type "http://mydomain.com" and it redirects to www.mydomain.com and it works. But if I type "https://mydomain.com" (with HTTPS instead of HTTP) it does not work. I presume that this is because whatever server is implementing the redirect (I use GoDaddy) doesn't have my SSL certificate so the connection can't be made.

I'm not sure how to resolve this problem. What I need, I think, is some web server that is on a fixed IP address and also has my SSL certificate, and can simply respond to all requests with the permanent redirect response. The only way I can think of to do this in AWS would be to set up an entire EC2 instance with my own web server, which is a lot of work and cost. Is there a better solution?

My company doesn't want to move our DNS or domain registration to AWS, so using something like Route 53 is probably not an option.

Thanks, Frank