What do you exactly want to do?
Do you want to manage these two domains separately? If not, you can simply create a domain alias.
If you want to make a copy of the site and manage them separately, you can
1. create a subdomain under the same user via [b]DH panel --> Domains --> Manage Domains --> Create New/Sub-domain
2. ssh to your new subdomain and copy all files from example.org to beta.example.org
If your site uses mysql database, you may want to create a copy of the database as well. The easiest way to make a copy of the database is using mysqldump