Critics have anxious that Moon's recent inter-Korean military deals would weaken South Korea's war readiness because the North's nuclear program remains largely intact.
It came after a historic meeting between Mr Kim and US President Donald Trump in Singapore in June in which the pair agreed in broad terms to work towards a nuclear-free Korean peninsula.
Extremely heavy fighting took place in October 1952 in a place known as "Arrow Head Hill".
President Moon Jae-in offers congratulatory remarks in a meeting with war veterans and military officials held at Cheong Wa Dae, Monday.
At Panmunjom, the Koreas are to spend 20 days clearing mines before withdrawing troops, weapons and guard posts in the area over the next five days.
Experts believe the South Korean and United States militaries had planted about 1 to 1.2 million landmines south of the DMZ, which measures 250 kilometers in length and 4 kilometers in width. The remains of a large number of Chinese and North Korean soldiers are also likely there.More news: The FBI Has Interviewed Kavanaugh Friend Mark Judge
The Joint Security Area is the only place along the demilitarized zone, or DMZ, where troops from the two Koreas face each other.
In recent years, South Korea has unilaterally removed thousands of mines annually near the DMZ as part of efforts to improve the lives and safety of civilians near the border area.
The two Koreas have agreed to disarm personnel at the JSA, which is patrolled by South Korean and USA troops, on one side, and North Korean soldiers, on the other.
"In parallel with the removal operation, we are going to proceed with the construction project of connecting roads between the two Koreas, with the aim of finishing it by the end of this year", the Defense Ministry said. The goal is make the area safe for troops from both North and South Korea.
A South Koren defence ministry spokesperson said the operation had begun on both sides, though there has been no confirmation by the North.
In April, South Korea said it had stopped broadcasting propaganda via loudspeakers along the border to "ease the military tension between the two Koreas". "We have to remove those mines, though we are not taking out all of the mines at the DMZ".More news: Tweet storm: Musk quits as chairman over false claims
"It's the start of peace", said Kim Ki-ho, head of the private Korea Mine Clearance Research Institute, according to CBS News.
At Arrow Head Hill, where some of the fiercest battles occurred during the Korean War, Seoul officials believe there are remains of about 300 South Korea and United Nations forces, along with an unspecified number of Chinese and North Korean remains.
The troops began removing mines on the southern part of the two sites. He also called for a stronger national defense, saying "peace can continue only when we have power and are confidant of protecting ourselves".
The South's Defense Ministry says the remains were found in North Korea during a 1996-2005 excavation project between the United States and North Korea.
Meanwhile, on Monday, South Korea held a ceremony marking the recent return of remains of 64 South Korean soldiers missing from the Korean War.
It is often used as a venue for talks between the two Koreas, including the two summits between Kim and Moon this summer.More news: Supreme Court Justice Kagan won’t discuss Kavanaugh