It is the second time in less than a week that the law – which was passed earlier this year – has been invoked.
The government said a post by the States Times Review (STR) contained “false statements of fact”.
But the STR editor, Alex Tan, said he would “not comply with any order from a foreign government”.
Mr Tan said he was an Australian citizen and had not received any “request from the Australian police to take down any article”.
Earlier this week, politician Brad Bowyer corrected a Facebook post after he was issued a “correction direction”.
What were the Facebook posts about?
The STR article claimed a whistleblower had been arrested for making claims about members of Singapore’s ruling party.
The government said no one had been arrested and also issued other corrections on its website.
“The STR…made scurrilous accusations against the elections department, the prime minister, and the election process in Singapore,” the government’s statement said.
Meanwhile on Monday, Mr. Bowyer – a lawyer and member of the political opposition group the Progress Singapore Party – was ordered to update a Facebook post he made on 13 November.
The updated post said, “this post contains false statements of fact, for the correct facts, click here”.
On the same day, Mr. Bowyer wrote a new post saying he was “not against being asked to make clarifications or corrections especially if it is in the public interest”.
But on Thursday, Mr. Bowyer clarified his earlier statement, saying: “Although I have no problems in following the law…that does not mean that I agree with the position they are taking or admit to any false statements on my part.”
He also said that, under the law, he must post the correction notice “regardless of whether I make an appeal”.
What is fake news law?
The fake news law, known as the Protection from Online Falsehoods and Manipulation bill, came into effect in October.
It allows the government to order online platforms to remove and correct what it deems to be false statements that are “against the public interest”.
A person found guilty of doing this in Singapore could be fined heavily and / or jailed for up to five years.
It also bans the use of fake accounts or bots to spread fake news – this carries penalties of up to S$1m (£563,000, $733,700) and a jail term of up to 10 years.
What has been said about it?
Critics said the fake news law threatens freedom of expression. Amnesty International said the law would “give authorities unchecked powers to clamp down on online views of which it disapproves”.
But Singapore’s law minister said free speech “should not be affected by this bill”, adding that it was aimed only at tackling “falsehoods, bots, trolls, and fake accounts”.
It has argued that the law safeguards against abuse of power by allowing judicial reviews of government orders.