If your existing contract has a fixed term, then by cancelling it you get by hit by early termination fees. If you're not in a fixed term contract, then presumably you're paying full whack now, and there's no big deal. If you are in a fixed term contract that should carry over.
From the T&Cs:
5. Where we have agreed with you to continue providing the same services that you received at your current address to your new address and this is during any minimum period, then that minimum period will continue at your new address (for example, where your services are on a 12 month minimum period, if you move during month 7 of your minimum period, the remaining 5 months of your minimum period will apply for those same services being provided at your new address). Where we have agreed with you to change the services you receive at your new address then a new minimum period may apply for those services and the service start date for that new minimum period will be the date that the equipment is installed at your new address.
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Where we have agreed with you to change the services you receive at your new address then a new minimum period may apply for those services and the service start date for that new minimum period will be the date that the equipment is installed at your new address.
If the minimum period is already over, and I do not change anything, there isn't a new minimum period, and I could just close the contract 1 month after moving (not that I'm planning to, you get the idea).
Thank you for coming back to us, if you have moved and just carried on with the previous contract to price change or new equipment then yes if the contract is up then indeed you can give your 30 days notice without any penalty.
So I've moved the contract without changing anything, the operator at the phone told me that there was just 1 month of minimum period for some reason, and instead I receive a brand new contract with a new minimum period of 18 months.
Good job, really. Now I'll have to cancel the contract and stay on the phone for hours listening to constant advertisements.
If it's a house move and you are moving your services over as is, without any changes there would not be a renewed contract term. The only thing that changes might be the account number since this is linked to the physical address. Were you able to get through to a member of the team?