Kim and Mr Pompeo agreed to arrange another summit between the Korean leader and US President Donald Trump “as soon as possible”.
The two officials also discussed the possibility of the US monitoring Pyongyang’s steps towards denuclearisation, the South Korean President’s office said on Sunday.
North Korean state media KCNA said: “Kim Jong-un expressed satisfaction over the productive and wonderful talks with Mike Pompeo at which mutual stands were fully understood and opinions exchanged.”
They also said the two sides agreed to hold a second round of negotiations between the US and North Korea as early as possible.
South Korean President Moon Jae-in said that Kim is expected to visit Russia soon and that Chinese leader Xi Jinping will also travel to North Korea, but did not give any further details.
Rodong Sinmun, North Korea’s state newspaper, featured eight photos from the meeting.
These photos included shots of Kim and Mr Pompeo smiling and shaking hands, as well as some with Kim’s sister Kim Yo Jong.
KCNA said: “Together with Mike Pompeo, Kim Jong-un appreciated the positively developing situation on the Korean peninsula, explained in detail the proposals for solving the denuclearisation issue and matter of concern of both sides and exchanged constructive opinions.”
During Mr Pompeo’s last visit to Pyongyang in July, he was denounced for making “gangster-like demands”.
The Secretary of State did not meet Kim on that trip.
Kim expressed enthusiasm for the future of talks between the two nations.
He said that dialogue would develop favourably “based on the deep confidence between the two leaders of the two countries”.
He also expressed gratitude towards Trump for making a sincere effort to implement the agreements that were made in their previous summit in June.
Mr Pompeo described his fourth trip to the North Korean capital as “another step forward to denuclearisation”.
He added he had a “good, productive conversation” with Kim, although he stroke a cautious note indicating that more work still needed to be done.
Mr Pompeo’s trip came after a stalemate between the two countries after North Korea resisted demands for the country to give up its nuclear arsenal.
In a statement, State Department spokeswoman Heather Nauert said Kim had invited inspectors to visit the Punggye-Ri test site in order to confirm that it had been dismantled.