In my ICloud settings an Apple ID is set which is not active any more. The e-Mail doesn't exist anymore and I have forgotten the PassWord to this Account/e-Mail. How can I sign this off and activate a new, valid Apple ID?
Thanks
Andreas
iPhone 5, iOS 9.3
Stop using the old Apple ID. You cannot remove it. Create a new Apple ID. You can do this in iTunes. Click on the Sign In link, then click on the link to Create a new Apple ID.
My understanding is that if you create a new Apple ID, then you will have to "rebuy" any purchases made with your old Apple ID. Purchases are permanently tied to the Apple ID you used to make them. I would exhaust all the possibilities to CHANGE you Apple ID first:
Change your Apple ID - Apple Support
If you have problems contact Apple Support at the link on the bottom of the page.
Apple ID
