Hi Bruno,
Do you have access to your Apache conf.d configuration files? If you do, just modify the one for your Moodle site and create an alias for both www.mydomain.org and mydomain.org to point to the same Moodle installation. For instance, on my Centos server I have multiple installations of Moodle. I have an alias set so that anyone trying to go to moodle or moodle23 will get my production site. For anyone to get to the others, they have to use a different address which will access a different configuration file and get a different alias pointing to the correct installation of Moodle.
In my moodle23.conf file I have the following:
Alias /mydomain/moodle23 /var/www/moodle23/web
Alias /mydomain/moodle /var/www/moodle23/web
You would need something like:
Alias /www.mydomain.org /var/www/yourmoodle/web
Alias /mydomain.org /var/www/yourmoodle/web
Hope this helps,
AL