If you add index.html to the first url, it loads. Is it possible you also have a .php .cgi or something in that folder? is the file name actually index.html in the RW project?
Prizes to you both. Very many thanks. Problem solved.
Scott was right - there was an old index.php file in the folder. God knows from where.
Webdeer was also right - the solution was to republish and re-upload. I had already cleared the server, and republished to a folder, ready to ftp, but of course, the php file was obviously lurking in the desktop folder, so when I uploaded, it went back up to the server. Lesson learned: clear everything before you republish all files.
Sometimes I think that I spend too much money on Rapidweaver, and too much time, but moments like this make it so worthwhile.