The man accused of murdering British backpacker Grace Millane makes his first appearance in the High Court at Auckland today.

Millane, 22, arrived in New Zealand in late November as part of a year-long OE. She went missing on December 2, her birthday, with family becoming worried after she did not respond to messages.

Police launched an investigation, and a week later officers found her body in a section of bush in the Waitākere Ranges.

A 26-year-old man was charged with her murder and appeared in the Auckland District Court on December 10, where he was granted interim name suppression, which had been opposed by police, the Millane family and the media.

Judge Evangelos Thomas had declined the application, however, the accused's lawyer Ian Brookie instantly appealed the decision, which automatically imposes a 20-working day suppression under New Zealand law.

However, since then there have been breaches by overseas media, hundreds of social media users and by search engine giant Google.

Justice Minister Andrew Little rebuked UK media for naming the murder accused, warning their actions risked jeopardising a fair trial, which could heap more misery on the grieving Millane family.

The naming of the man, which also occurred on social media, could be used to argue that he won't get a fair trial, Little said.

"The defence counsel will be looking for every opportunity to say fair trial rights might be compromised," Little said.

At today's court appearance, the murder-accused's interim name suppression is expected to continue, unless it timed out or his lawyer filed an appeal.

The British backpacker was last seen on CCTV on Saturday, December 1, entering CityLife Hotel in Auckland with the 26-year-old accused of her murder.

She had landed in New Zealand 10 days earlier and had been travelling around the North Island before settling in Auckland for a visit.