Then after DNS has changed you should change it to a 301 redirect from the old to the new.
It's a little more complicated since you set it up this way to start, but it can be done easily.
Delete the mirror.
Add the new domain, alternative-domain-name.org, as fully hosted, using the SAME ftp/shell user.
using shell or FTP rename the directories:
-- existing-domain-name.org -> TEMP
-- alternative-domain-name.org -> existing-domain-name.org
-- TEMP -> alternative-domain-name.org
delete hosting on existing-domain-name.org
Redirect existing-domain-name.org to alternative-domain-name.org via the panel.