Thank you, Ralf. I did not want to overwhelm people with any details but let's see now.
Could it be that your customer wants to use payment systems such as Moneybookers or Paypal, and did not express this very well?
Indeed, they offered to pay me using Paypal but I have not done this set-up for myself. Usually, I accept checks and that has worked fine for me. This time, the agency wants to pay me directly into my credit card. What card do I have? Just ordinary credit cards, like Visa or MasterCard. With a number and expiry date, nothing special about them. Maybe indeed it is safe to give it to them since I will not be authorizing any charges at all.
So these are all the details now.
Credit card number and expiry date are all you need to buy stuff, not to pay. In France at least, you can not be paid using your credit card number.
Get them to follow the usual procedures. Bank transfer or check (since you accept them). They left the payment options out on the contract and you are therefore entitled to decide how the payment should be done. If they want something different for next time, you can work it out, but this time, it's up to you.