Police appealed for public help to find the alleged blackmailer, at a press conference in Wellington Tuesday.
They revealed an investigation had been underway since November, when anonymous letters were sent to giant dairy cooperative Fonterra and a farmers' lobby group.
The letters were accompanied by small packages of milk powder containing a concentrated form of 1080 (sodium fluoroacetate), a poison used by New Zealand's Department of Conservation to control introduced pest species such as possums and rats.
The letters threatened to contaminate infant and other formula with 1080 unless New Zealand stopped using the poison for pest control by the end of March, police said in a statement. The letters said the threat would be made public if the demand was not met.
New Zealand's heavy use of 1080 -- it uses about 80% of the world's manufactured stocks, according the country's Environmental Protection Agency -- has been a controversial issue, with conservationists and farmers generally supporting the measure, and some hunters and animal rights activists opposed.
Primary Industries Minister Nathan Guy said the threat amounted to "eco-terrorism."
Prime Minister John Key said at a press conference that police had advised that there was a low likelihood of the threat being carried out.
Meanwhile, government officials said supplies of formula had not been contaminated, and urged consumers to continue using the products.
"We are confident that New Zealand infant and other formula is just as safe today as it was before this threat was made," said Scott Gallacher, Deputy Director-General of the Ministry for Primary Industries.
"People should keep using it as they always have."
He said more than 40,000 samples had been taken without finding a trace of 1080, and food security measures had also been boosted in response.
These included a strengthened security measures in retail stores and a boosted presence at manufacturing sites, and enhanced milk and product testing, including the introduction of a test for 1080.
"This criminal threat is designed to cause fear in order to generate a political outcome. It is using food as a vehicle but should not undermine confidence in our world-class food safety system or in any manufacturer," he said.
He urged consumers to check formula packaging for any signs of tampering as a result of the threat.
Appeal for help
Police said an Auckland-based investigation team was pursuing a number of lines of inquiry in locating the source of the threat, but the time was right to appeal to the public.
"Whilst there is a possibility that this threat is a hoax, we must treat the threat seriously and a priority investigation is underway," said Deputy Commissioner Mike Clement.
"You might be aware of someone who has strong views on the 1080 issue and made threats, or has discussed how to access supplies of 1080."
He also called on the alleged blackmailer to come forward.
"The letter writer may not have really considered the implications of their actions when this communication was drafted. Now is the time to put this right by picking up the phone and calling us."
Blackmail carries a maximum sentence of 14 years jail in New Zealand.
New Zealand is the world's leading dairy exporter, and dairy cooperative Fonterra is the country's biggest company, with revenue of $16.3 billion last year.
CEO Theo Spierings described the threat in a press conference as "a despicable crime."
The global diary giant has been beset by problems before, most recently in 2013, when it ordered a global recall of products -- including infant formula -- after fears they might contain bacteria that could cause botulism.
The fears proved unfounded, as the bacteria detected was not a botulism-causing strain.
New Zealand Infant Formula Exporters Association chairman Michael Barnett said eyes were now on the response to the threat in key markets such as China, the biggest consumer of New Zealand dairy exports.
"This could be extremely damaging for New Zealand's exports offshore," he told CNN affiliate TVNZ. "Our whole reputation as an exporter of food is at risk because of this nutter."
Trading in all New Zealand-listed dairy companies was halted Tuesday with the announcement of the threat.