Emails – send 2nd email on completed order

Hi,

I’m creating a small webshop for downloadable products with creditcard payment.

I’d like to automatically send a second email when an order is completed.
So the customer receives the customer-completed-order.php email and xxxx.php email. The second email can also be the customer-processing-order.php email, since it won’t be sent out anyway.

Is this possible ?

Thanks,