Praise is rolling in for Prime Minister Jacinda Ardern after her appearance on The Late Show with Stephen Colbert.

During the show, Ardern talked about being stopped for a chat while shopping for maternity bras, and the moment partner Clarke Gayford tripped over at a meeting with US President Donald Trump.

Ardern was an invited guest on the show, which has a viewership of four million. She was introduced by Colbert - a Hobbit fan - as the biggest thing to hit New Zealand since "Frodo dropped the ring into Mt Doom".

Some American viewers gushed that they wished she was from their country, or that they wanted to move to New Zealand.

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Her success on the show and the positive public response is similar to the reaction she's had while on formal United Nations duties in New York.

SCOTT KOWALCHYK/CBS Prime Minister Jacinda Ardern chats with The Late Show host Stephen Colbert.

Colbert introduced her as the youngest female world leader and second elected leader in history to have a baby.

And there were big "awww's" from the crowd when he showed the picture of Ardern holding baby Neve on her knee during a session of the UN.

Ardern joked that everyone was admiring her partner Clarke Gayford and not Neve.

Gayford was in the studio audience while baby Neve, who has attracted huge attention in the United States, was backstage.

Colbert raised the moment when world leaders laughed at Trump during his speech to the UN and noted that Trump seemed surprised.

"What did you experience?"

Ardern asked him: "Are you trying to create a diplomatic incident?"

SCOTT KOWALCHYK/CBS When Colbert suggested that Kiwis were "very vital and seem vigorous" Ardern said she'd never heard of us being described like that. "It makes us sound like a health product."

She told Colbert: "There was a little laugh and he said I didn't expect that response and there was a bigger laugh then people laughed with him."

"So no war between the US and NZ," Colbert responded.

Colbert also asked her about steel and aluminium tariffs imposed on New Zealand by the US, and Ardern told him about raising that with Trump.

But it was at that moment that Gayford knocked over a flagpole, she admitted.

"I raised it; then the flag went down ... as I turned around he was holding a long metal prong…

"I'm surprised no one leapt on it at that point because it looked like a weapon."

DON EMMERT/AFP Spoonley said Ardern and partner Clarke Gayford's decision to attend the UN with baby Neve was an example of New Zealand's relaxed attitude to ceremoney.

Colbert asked her about her handshake strategy and Ardern said she went for the "natural shake".

"I have quite a firm shake ... it's the way we shake in New Zealand."

When Colbert suggested that Kiwis were "very vital and seem vigorous" Ardern said she'd never heard of us being described like that.

"It makes us sound like a health product."

Colbert, who is a big Hobbit fan, also asked her about growing up near Hobbiton and Ardern said she found it "slightly offensive" that people think every New Zealander starred in Lord of the Rings or the Hobbit.

But when he asked if she had, she said: "Some of us auditioned but weren't successful ok?"

Before the show Ardern posted a Facebook live video from the green room to tell New Zealand about her day -she attended a trade meeting with Canada and Chile, and two speaking engagements.

Following the show Ardern said she would head back to her hotel to work on tomorrow's speech for the UN General Assembly.

"Which I haven't quite finished yet and I need to lock that down tonight."

THE PM IN NEW YORK

Ardern is in New York City for the United Nations General Assembly, and she's had a packed itinerary as she does the rounds.

The Big Apple has been welcoming to her, but she is on a charm offensive - she's been pushing for global co-operation on issues including climate change, inequality and trade.

Outside her UN engagements, she's been interviewed by the CNN's Christiane Amanpour, met Anne Hathaway and appeared on the Today show.

But it's her baby, Neve, who has really helped her steal the show.

Neve - and her dad, Ardern's partner, Clarke Gayford - were on the floor during Ardern's first speech to the UN, and Neve even spent some of it on her mum's lap.

Ardern hadn't expected Neve to be there, but a well-timed photograph from the occasion reveals her delight at getting to share the moment with her baby.

US media, diplomats and politicians have reacted to Neve's presence - and they're in favour.

Some of Ardern's New York schedule highlights have included a keynote address at the ninth-annual Social Good Summit, meeting with UN Secretary-General António Guterres, the keynote speech at opening of Climate Week 2018, a statement at the Nelson Mandela Peace Summit Opening Plenary and attending US President Donald Trump's annual reception for heads of delegation.

Her final two days - Wednesday and Thursday, US time - are just as packed, if not more so.

Ardern's last major appearance will be to deliver a national statement to the General Assembly.

The Late Show with Stephen Colbert will screen in New Zealand on Thursday night.

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