The North Korean foreign minister, Ri Yong-ho, said on Saturday his nation would never abandon its nuclear weapons if it could not trust the US.
Most flee via China, which has the longest border with North Korea and is easier to cross than the heavily protected Demilitarised Zone (DMZ) between the two Koreas.
'Why has President Trump given up so much?' Mr Trump said in a mock news anchor voice, before adding: 'I didn't give up anything'.
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U.S. Secretary of State Mike Pompeo met with Ri in NY and is planning to go to Pyongyang next month to continue discussions.
The canine gifts come after a September meeting between Moon and North Korea leader Kim Jong Un in Pyongyang, at which Kim agreed plans to shutter a missile-testing site and visit Seoul.
"The recent dramatic improvement of the North-South relations and the atmosphere of cooperation clearly show how decisive the role of trust-building can be", he said.
The pair met in-person for the first time in June for a historical Singapore summit which sought to officially commit Pyongyang to denuclearise the Korean peninsula.
"The perception that sanctions can bring us on our knees is a pipe dream of the people who are ignorant about us".
This includes at two UN Security Council (UNSC) meetings, chaired by Trump and U.S. Secretary of State Mike Pompeo respectively.
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"We must not forget what's brought us this far: the historic worldwide pressure campaign that this council has made possible through the sanctions that it imposed", Pompeo said.
Earlier Saturday, North Korea's foreign minister said that there was "no way we will denuclearize" without concessions being made by the US.
Pompeo said some of those sanctions have been violated, including through ship-to-ship transfers of refined petroleum to North Korea and continued hiring of North Korean laborers overseas. Post-summit nuclear negotiations between Washington and Pyongyang were rocky and quickly settled into a stalemate. "But now we're at the dawn of a new day".
Although not legally binding, the North might also see an end-of-war declaration as a way to create political momentum that would steer discussions toward diplomatic recognition, economic benefits and security concessions.
In its Sunday edition, the Rodong Sinmun, the organ of Pyongyang's ruling Workers' Party of Korea, also called for easing of sanctions on the country.
North Korea's foreign minister has warned that there is "no way" his country will disarm while the United States continues to enforce sanctions.
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'The sanctions will stay in place until denuclearization occurs'.