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While the best option is to close the account as that is the only way to break the chain with the ex without his approval you might end up in a catch 22. To close most credit accounts you must zero the balance. This means that the joint account that is already at zero should be closed immediately. The account used by the ex on the other hand depends on how the issuer is willing to handle it. Suspending charge privileges is the number one request that you want to implement. The issuer might even offer to automatically close it once it hits $0. For obvious reasons all of this should have been ironed out better once divorce proceedings were implemented. The whole idea of joint accounts is that the parties have some control of one another like family or business partners. The moment the spouse became the ex he should have been exed from the account.
Is there anyway the EX and the DW can 3-way call the company? Would he be willing to take responsibility for the debt in exchance of removing the DW from the account? Seem's like the 2 are on semi good terms if he said he'd "work on it"