Moving a PHPBB3 Forum from one domain to another

Assuming your still using the same host... just a diffferent domain
Step 1 - Edit cookie settings via ACP to that of the new domain name
Step 2 - Edit server settings via ACP to that of the new domain name

and you should be fine

if you are transferring hosts too then,
Step 1 - Copy over all html files of the web site and phpbb3 directory to the new domain name's folder via ftp
Step 2 - Copy over all the database and make sure it is the same name also same user for it.
Step 3 - Edit cookie settings via ACP to that of the new domain name
Step 4 - Edit server settings via ACP to that of the new domain name