"Grace's family have been advised of this development and they are devastated", Det Insp Beard added.
DI Beard said on Saturday: "Sadly the evidence we have gathered at this point of the inquiry has established that this is a homicide".
"We don't know what's happened to the property of hers that has been missing", he said.
Ms Millane, from Essex, was described as a "lovely, outgoing, fun-loving, family-orientated daughter" by her father, David Millane.
He added: "I know that the thoughts of all New Zealanders will be with Grace's family tonight".
The 26-year-old being questioned is the person of interest police mentioned yesterday.
Investigators had said yesterday that they held "grave fears" for Grace's safety.
"Several of her belongings, including her passport, are missing and police believe it is likely that Grace has these items with her".
"Dozens of calls have been made to the 0800 number and all of this information continues to be reviewed and assessed".
Beard said Millane had been out of contact with her parents for days and, especially worrying to her family, they weren't able to contact her on Sunday for her 22nd birthday.
Ms Millane, 22, from Essex, England, was last seen at an Auckland hotel almost a week ago.
New Zealand police confirmed on Saturday afternoon they are treating the investigation as a homicide.
Police have not uncovered any evidence the pair had met before.
"The issue here is Grace is still missing and we want to find her".
A vehicle of interest had been secured and police were trying to find out exactly where the vehicle had been since Millane's disappearance.
Before she vanished, she was staying at the Base Backpackers hostel in Auckland.
On Saturday police released images of a pink watch and a necklace belonging to her and asked for the public's help in finding them.
She graduated from the University of Lincoln in September and had embarked on a year-long worldwide trip.
That was especially so for David Millane, as police had met the distraught father, but also for other family in the United Kingdom, including Millane's mother Gillian, unable to travel to New Zealand because she was recovering from an operation.
David Millane, the father of missing backpacker Grace Millane, says his daughter has "never been out of contact for this amount of time". "We are all extremely upset, and it's very hard at this time to fully describe the range of emotions we are going through".
He said Millane's family had been told and they were "devastated".
He wouldn't speculate where Grace was killed, Beard said.
"There's family back in the United Kingdom and our hearts go out to them as well".