"How do I remove payment method in Apple Account?: How to remove payments to apple services"

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You need at least on Method of Payment associated to your Apple Account. If you were to cancel a credit card, then it would remain as your Method of Payment, and you would add a gift card to get credit. If you lose payment, you'd lose services, such as iCloud Drive.

Verify if this is a Method you can use:Payment Methods that you can Use with your Apple ID - 

Add your new method of payment, and then delete the current one. You need at least on method of payment in there.