Sure - MySql has tools to do that.
If you're using phpMyAdmin ( http://www.phpmyadmin.net/ ) then it's quite easy and straight forward. Just look for the export/import option.
If you have telnet access to the server then look for the mysqldump command - you can read more about it here:
http://www.mysql.com/doc/en/mysqldump.html
That page should also tell you how to import the database onto your new system.
Hope this helped.