Unfortunately, your friend’s situation is not that unusal. While it makes it easier to have the cooperation of the site designer, it is by no means necessary.
Does your friend have the username/password for the account hosting his site? If so, just login and ftp the site’s files to your machine. If not, you may have to save each page from the web, and be a little creative in deducing how it all works, possibly doing a little scripting replacements.
If your friend doesn’t have the control of his domain, then that is a bit different.
If your friend has control of his domain, the old designer is not needed; just grab the file from the web via wget or “save as”, arrange for new hosting, change the DNS for the domain to the new host’s servers, upload the site’s file, and you are done!