Welcome to the forum, vellocet.
If you go in to your cpanel and click the "Backup wizard", you could download a compressed file of your "Home directory".
Then on your local computer, decompress it using winzip etc.
After that, ftp all files and folders under the sub directory to the new location "public_hrml".
If the forum software needs changes in a config.php etc to know where the forum files are, don´t forget to change those lines in it.
This procedure is a bit time consuming, but when done, visitors will only see example.net instead of example.net/folder name.