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Sounds a bit off to me that you can't instigate a cancellation of service just because a bill is outstanding. Ntl bill in advance for many services so you would generally be in credit, except for any call charges you have made.
In any case, when someone is saying they won't be able to keep paying, that surely should be a trigger for a supplier to want to allow disconnection so they didn't incru a bad debt.
Ultimately your contract with ntl simply says you have to give one month's written notice. Put it in writing sending it by recorded delivery to the address on the back of the bill. Normally ntl will deal with this by phone, but it seems this time they are being pedantic and sticking to the letter of your terms and conditions.
The only time you cannot cancel is if you are within your 12 month contract period. Are you outside of that?