Alright, I changed the Site URL following the instructions here (under the section "Using a pre-existing subdirectory install")
Everything is working great, but the navigation is still pointing towards /wordpress/page_name, and the client wants to change that.
If I go into the permalink settings, all of the options have the /wordpress/ in front of them.
I've tried re-uploading the .htaccess file, but it doesn't seem to work. The code in the .htaccess is :
# BEGIN WordPress
<IfModule mod_rewrite.c>
RewriteEngine On
RewriteBase /
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule . /index.php [L]
</IfModule>
# END WordPress
Is there something I'm missing? Thanks in advance for the help.