I found the problem is the "Show cart" link in the "Add to Cart" Pop-up. My customers site runs on https, but the "Show cart" link in the Pop-up points to http://....
Never mind....I found the Problem.
"Enable SSL for sensitive areas (recommended)" in the VM configuration was not enabled.
BUT, in my view it's a very strange behaviour. If your whole site is running over SSL( Forced by the Joomla configuration). The above VM config option should not be allowed to override it back to http, if it is not enabled.
@Thomas Kuschel
Thanks! I did not test your code yet, but it lead my to the correct solution of my problem.