Ardern said more people would get sick – older New Zealanders are "by far" at the most risk.
"Today's death is a reminder of the fight we have on our hands."
Ardern again called on New Zealanders to stay home – "stay at home, break the chain and save lives".
The woman in her 70s died at Greymouth Hospital early this morning. Director-General of Health Ashley Bloomfield told reporters the woman who died early this morning was initially thought to have influenza.
He was still waiting to hear how the person who died had contracted the virus, but said the link was likely an overseas link.
The district health board has placed 21 staff who had treated the woman into self-isolation.
None of the staff treating the person are symptomatic of Covid-19, Bloomfield said.
Bloomfield said the person who died was "well known" to DHB staff, but when they were first admitted they did not know the patient had Covid-19.
Ardern said gloves and surgical masks were worn by staff treating the woman, but eye goggles were not worn initially.
Ardern said she has not been in contact with the dead woman's family as they had asked for privacy.
Bloomfield said New Zealand has sufficient hospital capacity to deal with the outbreak.
Nine patients in hospital, 514 total cases A total of 56 people have now recovered, while nine people are in hospital, Bloomfield said.
One of those patients is in ICU on a ventilator.
There have been an average of 1786 tests conducted per day.
There are now 514 cases in total in this country, with most cases still having strong links to overseas travel.
Asked how many cases are being treated as community transmission, Bloomfield said he was not able to say.
But he had "no doubt" the number of such cases would increase.
Bloomfield said 4 per cent of cases were Māori and 2.3 per cent were Pasifika people.
He called on those handling food to step up their precautions and reminded Kiwis to wash their hands as regularly as possible.
PM: 2000 calls about people flouting lockdown Ardern called out people online who are bullying those with Covid-19, or suspected to have the virus.
She said the people with Covid-19 "deserve our support".
Police have the power to arrest those who are not following the rules.
She said she was "frustrated" by reports of people not following the lockdown rules.
Some 2000 calls have been made about people breaching the lockdown restrictions.
Ardern said she would be speaking to Cabinet about whether or not supermarkets will be required to close over Easter's Good Friday and Easter Sunday.
She is seeking advice on supermarkets' rising prices during this time – she said she would provide an update on this tomorrow.
Ardern said she was aware of the issues around getting Halal meat to those who require it.
She said the Government was working on this.
Ardern said she worried that older New Zealanders were not asking for help if they need it.
But she asked them again to "stay home" and to reach out to friends, family and neighbours.
Ardern spoke directly to those over 70 who were not listening to their children who are urging them to stay home: "Please listen to me, stay home."
Prime Minister Jacinda Ardern. When it comes to people continuing to buy and sell goods using Facebook Marketplace, Ardern cautioned people against this.
The Ministry of Foreign Affairs and Trade was working on cases where New Zealanders were still suck overseas.
Repatriation in some parts of the world remained difficult. She said there are roughly 300 New Zealanders around the world on cruise ships.
Asked how many foreign nationals are in New Zealand, Ardern said she was not able to say.
Ardern said the Government is not willing to draw any conclusions from the fact that the number of new cases has been decreasing.
She said there is always a lag time with Covid-19 diagnosis.
Police website to dob in lockdown breaches Police have today unveiled a new online form the public can use to report lockdown breaches.
The online form can be found at www.105.police.govt.nz
The form can be used to report self-isolation breaches and businesses continuing to operate when they are not part of the essential workforce.
Police Commissioner Mike Bush urged people to use the online form rather than phoning police.
"We know that people want to do the right thing if they see people flouting the restrictions, but we want to ease the load on the non-emergency phone number.
"Police will take the information reported online and make contact to remind those breaching the restrictions of their responsibilities."
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On Friday, Prime Minister Jacinda Ardern said she expected the numbers to continue rising until April 6.
"Don't be disheartened when you continue to see our numbers increase," she told The Nation yesterday.
"I expect that we will have a number of cases and those cases will grow steeply because of the lag period of Covid-19."
Despite the consistent "stay home, stay local" message from the Government, some continue to break the rules.
In a Covid-19 update this morning, officials again attempted to clarify the lockdown rules.
"You should stay at home as much as possible, except for going for a walk or picking up essentials," it says.
So far, more than 615,000 cases of Covid 19 have been confirmed across 177 countries.
NeedToKnow3 The United States has become the epicentre of the global pandemic, with more than 100,000 confirmed infections and nearly 1600 deaths.
US President Donald Trump said he was considering placing a quarantine on New York as well as parts of New Jersey and Connecticut.
New York is the hardest-hit state in the United States so far, with more than 52,000 confirmed cases and at least 728 deaths.
Meanwhile, Spain has suffered another staggering surge in coronavirus deaths overnight as it struggles to fight back against the disease.
• Covid19.govt.nz: The Government's official Covid-19 advisory website
