This thread got me to experimenting. My site is at www.mysite.com/drupal
so I uncommented this in .htaccess
  # Modify the RewriteBase if you are using Drupal in a subdirectory or in a
  # VirtualDocumentRoot and the rewrite rules are not working properly.
  # For example if your site is at http://example.com/drupal uncomment and
  # modify the following line:
   RewriteBase /drupal

I then tried out both of these possibilities:

  # If your site can be accessed both with and without the 'www.' prefix, you
  # can use one of the following settings to redirect users to your preferred
  # URL, either WITH or WITHOUT the 'www.' prefix. Choose ONLY one option:
  #
  # To redirect all users to access the site WITH the 'www.' prefix,
  # (http://example.com/... will be redirected to http://www.example.com/...)
  # adapt and uncomment the following:
  # RewriteCond %{HTTP_HOST} ^example\.com$ [NC]
  # RewriteRule ^(.*)$ http://www.example.com/$1 [L,R=301]
  #
  # To redirect all users to access the site WITHOUT the 'www.' prefix,
  # (http://www.example.com/... will be redirected to http://example.com/...)
  # uncomment and adapt the following:
  # RewriteCond %{HTTP_HOST} ^www\.example\.com$ [NC]
  # RewriteRule ^(.*)$ http://example.com/$1 [L,R=301]

Now I can access the site with mysite.com and http://www.mysite.com
but not with www.mysite.com.

Now, of course, it could be our astaro firewall, which seems probable but just wanted to make sure:
The firewall says :
Astaro Security Gateway Version 8 http://www.astaro.com
Content blocked
While trying to retrieve the URL: http://www.dsnextgen.com/?a_id=115576&domainname=www.mysite.com
The content is blocked due to the following condition: The URL you have requested is blocked by Surf Protection. If you think this is wrong, please contact your administrator
Report: Malicious Sites not allowed.
Your cache administrator is:

Tony