United States officials say those steps can be easily reversed and have not been verified, and Washington has slapped additional economic sanctions on the impoverished country. Sanctions have also been strengthened since the meeting with Mr Trump.
Seoul and Washington are in a security alliance and the USA stations 28,500 troops in the South.
"It is the unwavering position of our party and the republic's government and my firm will that the two countries as declared in the June 12 joint statement. take steps to establish a permanent and stable peace regime and push toward the complete denuclearization of the Korean Peninsula", he said.
President Trump looking forward to another summit with Kim Jong Un; Rich Edson reports from the State Department.
Kim Jong Un has issued a stark warning to the USA over sanctions placed on North Korea, saying the country would have "no choice" but to act if Washington continues with its "one-sided demands".
North Korea is subject to various sets of United Nations Security Council sanctions related to its banned nuclear and ballistic missile weapons programmes.
The North Korean leader on Tuesday reiterated his determination to achieve complete denuclearization and also called for economic development in his annual New Year's address, stressing the need to strengthen the North's defense.
Although Pyongyang did not conduct nuclear or missile tests previous year, satellite images have pointed to continued activity at the North's related facilities.More news: Russian Federation : 4 dead in apartment collapse, apparent gas explosion
Kim said he was willing to resume without conditions joint projects with South Korea that were frozen due to political tensions and North Korea's weapon tests. Washington halted some large-scale military exercises with Seoul, although smaller drills continued.
"North Korea seems determined in 2019 to receive some sort of sanctions relief".
The televised speech is closely watched as a rare public appearance for the young leader, setting the tone for his domestic and foreign policies for the year ahead.
Duyeon Kim, from the Center for a New American Security, told CNN that the New Year's message was "confident" and "normal", but added that the message to the US was clear: "He still sent a very firm word of caution, bordering a nuanced threat, that if Washington doesn't keep its Singapore promise and continues with sanctions, then he has Plan B in mind and will go his separate way".
North Korea has not responded to Washington's calls for senior working-level talks to follow up on the June 12 summit, apparently preferring to engage Trump rather than other United States officials in its negotiations.
Kim delivered the address while sitting on a sofa in an office with portraits of his father, Kim Jong-il, and grandfather Kim Il-sung in the background. 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.
Dressed in a Western style suit, Kim spoke from a dark leather armchair in an office lined with packed bookshelves and paintings of his father, Kim Jong Il, and grandfather Kim Il Sung - a setting that resembled more intimate addresses from the White House by the USA president.
Much of Kim's speech Tuesday focussed on North Korea's moribund economy, saying that improving people's lives was his top priority and tackling energy shortages was an urgent task. Private analysts have accused North Korea of continuing nuclear and missile development, citing details from commercial satellite imagery.