If you are currently under contract you will need to pay an Early Termination Fee. The best time to cancel is at the end of a billing cycle. The account holder needs to call in with all the information on the account.
The issue and preferred resolution, in GetHuman818504's own words:
Leaving the state, want to cancel my account by the end of the week! I need my service cancelled by Friday.
I trust that CenturyLink will make this right and come up with a resolution that is fair.
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