I went to the phpBB homepage and downloaded the latest complete version and unzipped that to my local pc. Then I uploaded the whole thing, overwriting the existing online files. (Of course I did a backup first!)
Then I had to "chmod 664 config.php" so that the installer could run, and then I went to "mydomain.com/forum/install/install.php" and selected "upgrade", and filled in the database details on that page.
Once this was done, I closed the browser to get rid of any session data, and then also deleted the /config and /install directories as per instructions.
Start the browser again, and everything was working.
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