Police have not yet found the body of the 22-year-old from Wickford in Essex.
The suspect, who was being questioned by police in relation to her disappearance, will remain in police custody until Monday, when he is due to appear at Auckland District Court.
Ms Millane has been described as a “lovely, outgoing, fun-loving, family-orientated daughter” by her father, David, who has flown out to New Zealand.
The last confirmed sighting of Ms Millane was at 9.41pm on Saturday 1 December, at the Citylife Hotel in central Auckland, when she was seen with a “male companion”.
She is known to have used the Tinder dating app while in New Zealand.
She had graduated from the University of Lincoln in September and was on a year-long round the world trip.
After visiting Peru, she arrived in New Zealand alone on 20 November and had been in frequent contact with her family up until 1 December – the day before her 22nd birthday.
She was staying at the Base Backpackers hostel in central Auckland prior to her disappearance.
Investigators announced they were treating the case as murder having previously expressed “grave fears” for the safety of Ms Millane.
Detective Inspector Scott Beard from Auckland Police said: “Grace’s family have been advised of this development and are devastated.”
Earlier, police had released images of Ms Millane’s watch and necklace and also revealed that her passport was missing.