Credit Card Payment - Legal Question

[MJDale]

On 16 March I had some blinds fitted. The sales rep came to measure up and took a deposit by credit card. This amount has appeared on my credit card statement.
When the sales rep came back to fit the blinds he took the outstanding amount, also by credit card. This amount has not yet appeared on my statement.
I have now been contacted by the blind company who state, more or less, that they have lost my credit card details and that they have therefore been unable to take the payment owed. Despite the fact that this is a mistake entirely on their part, the letter is written in tones which are far from apologetic and the blind company demand that I send them payment as soon as possible.
How do I stand, legally?
Whilst I do not want to end up in the small claims court I feel that I have complied with my side of the contract by giving my credit card number and signing on the line.

[Sensei]

You received goods which you did not pay for. All those other details aside, that's the bottom line and you are responsible for payment. If the payment was lost in the mail, would you expect them to just give up saying, "Oh well, (s)he tried to pay, but it wasn't his fault."???