The latest talks began at the border truce village of Panmunjom on Sunday, the South's Yonhap news agency said, citing a Seoul official.
Thursday's delay was widely reported as a snub by North Korea, which has already kept the US waiting for weeks since President Donald Trump agreed with Chairman Kim Jong-un on June 12 that America's war dead would be recovered, including the "immediate repatriation" of those already identified by the North.
Returning the remains of the US soldiers who perished during the 1950-53 conflict was part of a deal signed by the North's leader, Chairman Kim Jong Un, and US President Donald Trump during their landmark summit in Singapore last month.
The North is believed to have about 200 sets of remains ready to be transferred, and U.S. Forces Korea has moved 100 caskets to the border area in anticipation that they will be turned over.
The last amnesty was offered in August 2015, when North Korea celebrated the 70th anniversary of the Korean Peninsula's liberation from Japanese colonization and the 70th anniversary of the founding of the Workers' Party of Korea. More than 36,500 USA troops died in the conflict.
The meeting was the first between a US and North Korean general since March 2009.
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"I firmly believe that the strong will, honest efforts and unique approach of myself and Your Excellency Mr. President aimed at opening up a new future between the DPRK and the US will surely come to fruition", the letter reads. State media say that Kim has harshly reprimanded local officials over a delayed construction project.
Graham then had a message for Pyongyang, referencing the fact that Kim Yong Chol, a top adviser to Kim Jong-un and a former spy chief, had asked Pompeo if he slept well during his visit to North Korea. Sources say that North Korea and the United States may hold working group meetings as a follow-up to the July 15 meeting.
The U.S. government estimates that the North Korean government has about 80,000 to 120,000 political prisoners in detention.
US and South Korean officials say they are ready to help North Korea revive its economy if it gives up its nuclear program.
On Monday President Trump will hold bilateral meetings with Russian President Vladimir Putin. The establishment media and their military industrial complex backers don't want to see peace in Korea, but even those who don't like Trump but manage to remain objective can see progress.
The Pentagon has said North Korean officials have indicated in the past they have the remains of as many as 200 US troops.