Auckland Police Detective Inspector Scott Beard said a 26-year-old man is being questioned in relation to Grace Millane's whereabouts.
Miss Millane's body has not yet been found, but Det Insp Beard said police were "determined to do all we can to find her".
"He was brought back here to the Auckland Central police station and is now speaking to us in relation to the murder of Grace Millane".
He added: "Today were have identified and secured a vehicle of interest, and we are now making enquiries to find out exactly where that vehicle has been".
The last confirmed sighting of Millane was at 9.41pm the same day at the Citylife Hotel in central Auckland, when she was seen with a "male companion".
He said Millane's family had been told and they were "devastated".
Investigators had said yesterday that they held "grave fears" for Grace's safety.
Speaking at a news conference in Auckland on Friday, Grace's father, David Millane, said his daughter had been "bombarding us with numerous photographs and messages of her adventures" before she went missing.
Images of her pale pink watch and silver necklace were released by police after it was noted they were missing from her belongings at the backpackers' hostel where she was staying. Local police were eager to find her and return her to her family, but new evidence has changed the course of the case.
He said: 'Just after 3pm this afternoon (Saturday) a 26-year-old man was located at a central Auckland address by police.
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Asked why police were now treating Millane's disappearance as a murder, Beard said it was a combination of factors.
The much loved daughter and sister arrived in New Zealand on November 20, as part of a year-long, worldwide overseas experience that began earlier this year in South America.
Det Insp Beard added: "We still do not know where Grace is".
Asked if the pair had met on the Tinder dating app, the detective said: "I'm not going to speculate on how they met".
Millane's father David, a millionaire property developer, flew into New Zealand early yesterday.
"The issue here is Grace is still missing and we want to find her".
"I would like to take this opportunity to appeal to anybody who has seen, spoken to, come into contact with Grace over the last few days and come forward with any detail, no matter how small", he said.
Grace last made contact with her family last Saturday, December 1.
He tearfully explained that his daughter has never been out of contact with her family for such a long period of time, usually contacting them on social media.
They are trawling through hours of CCTV footage to trace her movements.
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They "seem to show a great deal of empathy towards some of the sad stories that I read with them", Mr Cox said. His daughter understood that her bullying was the reason she was forced to walk, Cox said.