No luck so far finding an answer to this in the help documentation.
Is there a way of CC'ing a copy of each welcome message to a specific email address?
CCing a copy of each sent welcome email isn't possible. What is it that you're trying to accomplish with that? There might be a different way you can accomplish it.
One of our customers would like to receive a notification each time a new member subscribes...
Your best approach in that case would be to do some custom code on your website (using PHP or whatever language you have available), so that the form is emailed to your client before being submitted to Campaign Monitor.
It does require some web development skill, but isn't too complicated for an experienced dev.
Hey Sting Tech
Another solution could also be that your customer, signed up with the rss feed to the specific subscriber list, your customer wants to monitor.