her estate is liable for the loss
if she left no will then you would be the inheritor of her estate and you would get everything after the debts have been paid
I think there is a general ( but mistaken ) feeling that the assets go to the inheritors and the debts sort of die with the person - but one is offset against the other and then the remainder ( if an asset ) goes to the heir
If you are gonna go for the idea of "my wife's only asset is a credit card debt of £680" I think you should seek professional advice - kick off with the good old CAB there will be one local to you
I don't know the answer to your question, but I did get enormous help and support from Answerbank members when my husband died about eighteen months ago.
Take your time to sort things out, and ask again anytime.
You are not liable personally but her estate is. If you are the executor then the debt will have to be repaid from the estate, providing there is enough remaining after prior obligations have been settled.
just as a matter of interest - if the deceased has a number of debts - in what order are the creditors paid , if there is not sufficient funds in the estate to pay them all ?