How wrong I was. 1and1 technical support came through!
I needed to change the .htaccess file in the root directory to the following:
--------------------------
# BEGIN WordPress
<IfModule mod_rewrite.c>
RewriteEngine On
RewriteBase /
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule . /index.php [L]
</IfModule>
# END WordPress
--------------------------
... and it worked. Who would have ever known?
Thank you for your help, t-p!