"If the USA does not keep its promise made in front of the whole world. and insists on sanctions and pressures on our republic", Mr Kim said, "we may be left with no choice but to consider a new way to safeguard our sovereignty and interests".
They met another two times after that but the most historic summit of 2018 was the North Korean leader's meeting with US President Donald Trump in Singapore in June.
"I am willing to sit with the United States president any time in the future and will strive to produce outcomes that would be welcomed by the global community", Kim said, wearing a suit and tie and seated in a plush leather chair overlooked by paintings of his father and grandfather at work.
"[But] we could be left with no choice but to seek a new way if the USA does not make good on its promises, misjudges our patience, while seeking to force things unilaterally and clinging to sanctions and pressure", Kim noted.
North Koreans' only had positive things to say about the New Year's speech.
The North Korean leader, who had been described by Trump as a "Little Rocket Man" with nuclear ambitions, humanized himself by taking a late night stroll on Singapore's modern streets the night before the summit that set off a social media firestorm.
Kim also called for South Korea to "completely stop" joint military drills with the United States involving strategic assets, while multilateral negotiations should be pursued to build a permanent peace regime on the Korean Peninsula.
Although Pyongyang did not conduct nuclear or missile tests previous year, satellite images have pointed to continued activity at the North's related facilities.
There are also plans for Kim Jong-un to travel to the South's capital Seoul for another inter-Korean summit but again, those plans have not been confirmed yet.
He added: "I also look forward to meeting with Chairman Kim who realizes so well that North Korea possesses great economic potential!"
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Clad in a suit and tie and sitting on a plush armchair, Mr Kim said bilateral ties would propel ahead if the U.S. were to "respond by taking trustworthy and corresponding, practical action". He also called for a halt to the deployment of "strategic assets" on the Korean Peninsula, an apparent reference to USA bombers and submarines capable of carrying nuclear weapons.
The North Korean leader is asking Washington to take the next step to break the stalemate, such as allowing the reopening of the joint Kaesong Industrial Complex, analysts said.
He spoke warmly of his three 2018 meetings with South Korean President Moon Jae-in and of the rapprochement between the two nations, but he said that progress should be consolidated by ending joint military exercises with the United States.
In the 30-minute speech, Kim spent more than 20 minutes highlighting his campaign to create a "self-reliant" economy despite a "harsh economic blockade".
The annual New Year's address is a tradition Mr Kim picked up from his grandfather, founder of the communist country, Kim Il-sung.
The North Korean leader is also indirectly demanding his neighbors to the South push the U.S. and U.N Security Council harder to lift sanctions in exchange for the two countries to resume inter-Korean cooperative projects, Cheong said.
But North Korea for decades has been pushing a concept of denuclearization that bears no resemblance to the American definition, with Pyongyang vowing to pursue nuclear development until the United States removes its troops and the nuclear umbrella defending South Korea and Japan.
To that end, Kim used his New Year's address to call for a permanent end to joint military drills between the US and South Korea.
Mr Kim also met three times with Chinese President Xi Jinping, which boosted his leverage by reintroducing Beijing - Pyongyang's main ally - as a major player in the diplomatic process to resolve the nuclear standoff.
It is thought the North Koreans will push for a quick second summit, as they like their chances of winning concessions from Mr Trump that they might not get from lower-level officials.