Commerce Minister Simon Power has previously said internet termination would only be introduced if the new laws prove ineffective at curbing illegal downloading.
The power to cut off internet access will sit with the District Court, rather than the Copyright Tribunal.
However Curran said can not work in the long term.
Announcing Labour's policy on digital copyright today, Curran said Labour will also undertake a review of the Copyright Act, with the aim of introducing a new bill within 18 months to update and extend the framework for digital copyright in New Zealand.
"Termination is unsustainable," Curran said.
"Labour voted for the bill in April because we stuck by a commitment to work with the Government to enable internet service providers and rights holders to reach a compromise on copyright law.
"That compromise meant that termination of internet access as an ultimate penalty for repeat copyright infringement remained in the bill, but could not be enacted without the consent of the Minister, but it is clear that this won't work long-term.
"Labour remains committed to protecting the rights of the creators of works," Clare Curran said.
"This is a debate about shifting power, access to information, out-dated business models and the immense potential of the Internet to change our world. No parliament anywhere knows what to do about it yet, but Labour is committed to trying to find solutions."