Source: docs.joomla.org
Many times you install Joomla in a sub-directory and then want to move it to a higher level directory, here's a short tutorial on how to do it.
Say you have installed Joomla in the following folder: public_html/tryjoomla. Now that you are satisfied with the site, you'll want to move to public_html.
1. Move all the files from the sub-directory (i.e., public_html/tryjoomla) to the upper level directory (i.e., public_html). You can use your favourite FTP client or the control panel that your hosting service provides.
2. Download and open the configuration.php file in a text editor.
3. Simply remove the tryjoomla folder name from the path. Look for the following lines
var $live_site = ''; var $log_path = '/home/username/public_html/tryjoomla/logs'; var $tmp_path = '/home/username/public_html/tryjoomla/tmp'; var $ftp_root = 'public_html/tryjoomla';
Change to:
var $live_site = ''; var $log_path = '/home/username/public_html/logs'; var $tmp_path = '/home/username/public_html/tmp'; var $ftp_root = 'public_html';
N.B. The $live_site variable rarely needs to be given a value. But if it was given a value during installation then edit that path as well.
var $live_site = 'http://www.example.com/tryjoomla';
Change to:
var $live_site = 'http://www.example.com';
4. Check your .htaccess. The subfolder should be removed there as well.
If you have cache enabled, login to the administrator backend (which will now be at http://www.example.com/administrator and nothttp://www.example.com/tryjoomla/administrator). Go to Tools -> Clean Cache and delete all cache files.
Source: forum.joomla.org
I haven't tested it yet (as you can see) but it's the only solution I found that doesn't include installation of an extra Component:
Update: Tested and working perfectly. Be sure to search for the line in step 3 thoroughly in case the copy paste doesn't work.