Based on the info you have provided, you don’t have a balance. The $0.68 is interest that was incurred on your account during the most recent monthly cycle before you paid off your final balance.
If your new balance is “$0.00,” then you have already paid the 68 cents when you paid off the card.
When you have paid your card down to a zero balance, they will send a statement for that last month so that you have a record of your payoff of the balance. It would be extremely odd for them to send a statement on a card with a zero balance at the start of the monthly cycle, absolutely no activity during the cycle, and a zero balance at the end of the cycle.
Is this a reputable card from a reputable company (like one of the big national banks)? If it is, then the only thing you should have to worry about is if there is an annual fee. You want to make sure the card is definitely closed in that case. If you do get charged an annual fee, you will have to call or write them again and explain that you were told the card was closed and that you want the fee removed and the card closed. If they will not do that, you can file a complaint with your state’s attorney general. Every time I have done that, I have been quickly contacted by the company in an effort to resolve the issue.
If it is a big reputable bank, they may make an error but will correct it if you bring it to their attention.
If you had poor credit when you got the card and got it from a lower tier outfit, there is no end to the possible bullshit fees and traps that may be in the small print. You will simply have to be persistent in making sure the card is closed. Like others have said, doing so in writing and using certified mail will make your case stronger if you are worried that they are not going to actually close the account.
Do not use the card again even for just a small, one-time emergency, or they may consider that an indication that you want to keep the account open and render all of your efforts to close it moot.