While a primary cardholder is alive, an authorized user is free to run up bills on that credit card account. After that cardholder dies, it's a different story. If you keep using the card, you could end up liable for the deceased's debt as well as your own -- and possibly face legal trouble as well.
Executors and Cards
When someone dies, an executor wraps up her estate. Part of the executor's job is notifying people of the death, settling outstanding debts and closing accounts. When dealing with a credit card account, an executor should cut up all cards and pay off any remaining credit-card debt out of the estate's assets, if the estate has sufficient funds. Discover, for example, says the executor should contact the company's Deceased Account Services. The company will then verify the death and close the account.
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Probate isn't fast, and executors sometimes make mistakes. If you're an authorized user on the deceased's card, it's possible you'll be able to ring up charges on it for a while. Unfortunately, that's illegal, as the contract was between the card company and the primary holder. Using the card after the primary card-holder's death is legally no different from using a stranger's card you found on the street. In addition, if you know the estate doesn't have the money to pay off the bills you ring up, you're committing fraud and could face civil and criminal charges.
If you co-signed a joint credit card agreement along with the deceased, the card legally belongs to both of you, even if she actually paid all the bills. You can keep using the card, but you're responsible for paying it off. If the co-signer died with a balance on the account, the estate should cover the debt, but the card company can legally come after you for the money if that's not the case.
Debt and Responsibility
Using the decedent's credit card after death when you have no right to do so usually has no good outcome. Even if the estate pays off the bills you ring up, that's going to eat away at the value of your inheritance. The safest move is to give your card to the executor and let him shred it.