WorkSafe, New Zealand's primary regulator for workplace related incidents, has opened a health and safety investigation, Ardern said.
She also announced a NZ$5 million ($3.2 million) fund to help small businesses affected by the eruption, after New Zealanders held a minute of silence to honour the victims a week on from the tragedy.
The official death toll from the surprise eruption on White Island, also known by its Maori name of Whakaari, stands at 16.
Two people whose bodies are believed to be in the waters around the island are still officially listed as missing.
A further 26 people remain in hospitals in New Zealand and Australia, many in critical condition with severe burn injuries.
Australian Foreign Minister Marise Payne met with Ardern in Wellington on Monday to express Australia's thanks to emergency and medical crews.
There has been growing criticism that people were allowed on the island, a popular destination for day-trippers, given the risks of an active volcano.
That has led to speculation the tragedy could foretell major changes for New Zealand's thrillseeker tourism economy.