You wait. How long? Could be a few months to a few years. Here's what I think is going on.
First, understand that no matter what you do, this unpaid debt is on your credit report and will remain there for 3-7 years. There isn't a damn thing you can do about that either. Even paying it in full will not make your bad payment history (or lack of payment history) go away. It's just not gonna happen because you can't change history.
The credit card company has sold your debt to a third party collection agency. If they were lucky, the credit card company sold it for ten cents on the dollar. The next thing that happens is the buyer of your debt will be calling or sending you a letter demanding payment in full, which may also include exhorbant "collection fees" they've added on.
Now I've helped a number of folks in your situation and not all outcomes were the same. A friend of mine got a letter from one collection agency demanding payment in full and threatening legal action if not paid by such-and-such a date. They bought the letter to me for advice. So with my friend present, I called the number provided in the letter and spoke with a "representative". In my conversation I did would a could to come across to them as "legal counsel" without coming out and saying I was a lawyer or anything. If I did that, I would be lying and two lies don't make a truth.
I explained that I and my "client" were well aware they could not take legal action of unsecured debt and if they continued to pursue the matter in this fashion then "we" would take legal action. Now the conversation was non-threatening and as professional as possible. After giving the impression (without coming out and saying it outright) that the debt would not be paid, the telephone conversation was "amicably" terminated.
About 3 months later they got another letter from a different collection agency, indicating that most likely the first agency had sold it to another. Now this 2nd letter was non-threatening and very professional. That letter indicated they were willing to "settle" for 25 cents on the dollar of the original $8,000 debt. That would be $2,000. I advised my friend to take that deal "provided" the agency would send a letter stating that this debt would be considered "paid in full" and permanently settled upon receipt of a cashier's check for $2K.
The company provided the requested letter and I went with my friend to the bank to get a $2K cashier's check, and then to the post office to mail it to them certified mail, return receipt required.
Now at tax time my friend received a 1099-C - Forgiveness of Debt tax reporting document for the $6,000 unpaid difference. This is common and perfectly legal. Additionally, it was reported as a "write-off" on his credit report. The tax on that $6,000 came to around $1,400 and it was paid. So for a total of $3,400 an $8,000 debt was settled.
If the credit card provider will not accept payment (according to what you say) then they're already in collections. Without knowing what collection agency has them, there's no one to contact. All you can do at this point is wait. That wait won't be very long either.