The sentence requires a conditional clause meaning 'if not' or 'except when'. 'Unless' means 'if not' and is the appropriate conjunction here: the speaker will pay only on the condition that the bill is sent again. 'Till' and 'until' indicate time, suggesting the action will happen after the bill is sent (which is logically possible but slightly awkward with 'not pay'). 'If' would require 'doesn't' to convey the conditional negative meaning. 'Unless' captures the intended meaning most precisely.
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