Despite testing negative Cave said the woman remains in isolation.
Two other prisoners were subsequently isolated after being in contact with the newly received prisoner.
"We have a duty of care to the men and women we manage in prisons, and to our staff, and everything we have put in place to manage the threat of Covid-19 has been based on prioritising their safety and wellbeing," Cave said.
Earlier today, a prisoner at Mt Eden Prison recently tested positive for covid-19.
Corrections Association vice president Paul Dennehy said the prisoner was one of 13 people remanded at Mt Eden and tested for Covid-19 shortly after arriving.
All but one of those prisoners at the central Auckland facility tested negative.
The prisoner had been in the Tikapa Moana/Firth of Thames area, but inside Counties Manukau's district health board boundary, before he arrived in metropolitan Auckland.
Director of public health Dr Caroline McElnay confirmed the prisoner had travelled from Counties Manukau.
At today's 1pm Government press conference, McElnay said four police officers, five Corrections staff and six prisoners were being treated as contacts.
The man who tested positive had been travelling with another person who was now isolating.
There was no suggestion the prisoner had travelled from outside any area currently in alert level 4.
Prime Minister Jacinda Ardern praised health and safety measures at the prison.
"I do want to acknowledge the protocols that are extraordinarily rigorous that have led to this detection," she said this afternoon.
She said newly arrived prisoners were supposed to wear masks, and were kept separate from other prisoners.
