Can we stop a customer's billing moving forward without canceling their subscription?

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Yes, there are several ways to handle this specific use case that we could recommend. The first would be to simply edit this customer's individual subscription and change the price to $0. With that change being made, even if the customer decides to remove their stored billing information, the subscription will remain active as all future invoices will automatically be generated with a $0 due amount and close.

Another way to handle this would be through the use of a coupon. If you create a new coupon and have the discount type set to percentage and 100%, you can then apply the coupon to the customer's account/subscription and have it begin discounting their new invoices moving forward. Coupons are good if you only want to temporarily stop the billings from occurring with this customer's subscription.