While Mr Ri spoke of a growing mistrust with the US, Mr Trump told supporters at a rally in West Virginia that he "fell in love" with Mr Kim during their historic summit in Singapore.
Ri says it's a "pipe dream" that continued sanctions and U.S. objection to a declaration ending the Korean War will ever bring the North to its knees.
North Korea's foreign minister has urged the United States to move past what he called seven decades of entrenched hostility if the countries' stalled negotiations are to restart.
"I want to reiterate that the future can be very bright for North Korea if it makes good on its commitment to final, fully verified denuclearization".
"The primary task for effectively implementing the DPRK-U.S. joint statement should be bringing down the barrier of mistrust", he added.
In recent weeks, Kim Jong-Un has said he would permanently dismantle North Korea's main nuclear complex, but only if the United States takes unspecified corresponding measures.
U.S. Secretary of State Mike Pompeo met with Ri in NY and is planning to go to Pyongyang next month to continue discussions.
China and Russian Federation have said the U.N. Security Council should reward Pyongyang for steps taken after U.S. Trump and Kim met in June and Kim pledged to work toward denuclearization.More news: Trump Orders Limited FBI Investigation To Supplement Kavanaugh Background Check
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'He wrote me lovely letters.
He said he and Mr. Kim have figured out they like each other, and they are planning a second summit, at Mr. Kim's request. "And they're great letters".
The Trump administration has been trying to revive nuclear talks with Kim Jong Un, the North's leader.
Nevertheless, Trump cheered his summit, claiming that North Korea "no longer" posed a nuclear threat.
Moon told an event in NY on the sidelines of the United Nations meeting that declaring a formal end to the 1950-53 Korean War would encourage North Korea to move further with denuclearization.
In August President Trump accused North Korea's ally China of undermining progress on denuclearisation because of its trade dispute with the US. He left Pyongyang in July saying that progress had been made, only for North Korea within hours to denounce him for making "gangster-like demands".
Last year, the Security Council unanimously adopted multiple sanctions resolutions in response to North Korea's nuclear and ballistic missile tests to choke off sources of revenue for the regime.
Kim had earlier compared comments by Trump to the bark of a "rabid dog", and Trump derided Kim as a "sick puppy" - before the apparent outbreak of puppy love.