Evidence they had found had indicated Grace was no longer alive, Beard said.
A 26-year-old man is being held in connection with her murder and a vehicle of interest has been found.
Detective Inspector Scott Beard briefed media in Auckland earlier this afternoon and said they were treating her disappearance as a homicide.
Ms Millane left the United Kingdom six weeks ago to travel the world after finishing university.
"A 26-year-old man was located at a central Auckland address by police".
Police had identified the man by yesterday and spoken to him before he was taken to Auckland Central police station today.
'Sadly the evidence we have gathered to this point of the inquiry has established that this is a homicide.
"We still do not know where Grace is".
The 22-year-old from Essex - described by her father as "lovely" and "family-orientated" - was last seen in New Zealand a week ago.
Earlier in the day police released photographs of a necklace and a watch in the hopes of finding her whereabouts.
Millane called his daughter a "fun-loving, family-oriented daughter" and urged anyone with information to contact police.
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A 26-year-old man, who was with Ms Millane when she was last seen, was speaking with authorities and police advised on Saturday afternoon that he would be charged with her murder.
Auckland City Police Detective Inspector Scott Beard said: "Detectives have assessed Grace's belongings that were found at the backpackers where she was staying". He also said investigators were looking into her credit card activity to see the last time she made a transaction.
The British backpacker, who graduated from the University of Lincoln in September, had been travelling alone in New Zealand for two weeks.
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.
Her father, David Millane, said they last had contact on the day she vanished.
They are trawling through hours of CCTV footage to trace her movements.
"All I can say is that we know they were together from the time at the Sky City Hotel".
He added that Ms Millane's family were "devastated" by the development and said: "We still do not know where Grace is".
"We've spoken to David [Grace's father] we've met him, but there is family in the United Kingdom and our hearts go out to them as well".
They still wanted to hear from anyone who had seen any of the items, or the black dress and white sneakers Millane was wearing when she was last seen.