The infected staff member had immediately isolated and been tested after a person in their household tested positive for the virus the day before, she said.
"Following notification … we took immediate action and began contact tracing as a precaution."
Fifty-five prisoners and 21 staff were identified as having had contact with the person while they were at work.
Staff were asked to isolate, get tested and follow regional public health guidance.
Prisoners were moved into separation units for isolation, and have been tested for Covid-19, Cave said.
"They will be cared for by Corrections' Health Services and custodial staff."
The prison was currently at Covid-19 "protection stage 2", which meant no face-to-face visits for private and legal visitors took place during the period the infected staff member was on site.
Prisoners were being supported, Cave said.
"We appreciate that this may be distressing for women accommodated within the prison and staff have been reassuring them frequently and distributing information sheets so that they know what has happened, and what they need to do to prevent Covid-19 from spreading.
"I would like to assure the whānau and friends of women at the prison that all of the women are safe and well, and the Christmas meal and activities planned for today are continuing – with additional physical distancing and PPE use."
Welfare support was also in place for staff. About 330 people work at the facility.
"I would like to thank our team at [the prison] for their professionalism and willingness to continue to carry out their essential work.
"Many people have given up time with their own families today to be on site, and we are extremely grateful for their service."
Preparations for Covid-19 had been in place since March 2020, including infection prevention and control plans, with detailed record keeping for both staff and visitors to sites to enable fast and thorough contact tracing – including the use of proximity detection technology to assist with contact tracing, Cave said.
Health screening for every person who entered the prison also took place. New prisoners were also kept separate from the existing prison population for their first 14 days behind bars.
"We are well prepared for this situation."
People in prison are some of the most susceptible to Covid-19, as many are vulnerable because of existing health conditions, and the ease of transmission the virus could have in the prison environment.
"Internationally, we have seen the devastating impact of Covid-19 in prison environments, with many prison staff and prisoners becoming seriously unwell, and a number of deaths occurring."