The two leaders had "agreed in principle" to strengthen Korea's defences in order to "counteract the provocations and threats of North Korea and to revise the missile guidelines to the level desired by [South Korea]", a statement said.
Trump has also spoken twice this week with Japanese Prime Minister Shinzo Abe.
President Donald Trump and South Korean President Moon Jae-in shake hands in the Rose Garden of the White House on June 30.
North Korea state television said in the afternoon that Pyongyang had tested a hydrogen bomb, and that the test was a "complete success". -South Korean military drills.
"North Korea has committed an absurd strategic mistake that will further accelerate its isolation from the worldwide community through its repeated provocations, such as launching intercontinental ballistic missiles (ICBM) and staging a nuclear test that not only heightened tension on the Korean Peninsula but also greatly undermined global peace", he was quoted as saying.
However, Trump said after the latest missile test that negotiations with Pyongyang were "not the answer".
It also objected to the translated title of the South Korean edition as "Capitalist People's Republic of Korea" and the book's cover that replaced the red star in North Korea's official seal with the USA dollar mark.
The U.S. Chamber of Commerce said in an email to members that it and other business groups "have received multiple reports" that the Trump administration is prepared to notify South Korea of its intent to withdraw from KORUS on Tuesday, and possibly sooner.
Under third-generation leader Kim, North Korea has been pursuing a nuclear device small and light enough to fit on a long-range ballistic missile, without affecting its range and making it capable of surviving re-entry.
The latest nuclear test comes amid heightened regional tension following Pyongyang's two tests of intercontinental ballistic missiles (ICBM) in July that potentially could fly about 10 000 km (6 200 miles), putting many parts of the mainland United States within range.
The US now has around 28,500 soldiers stationed in South Korea, and Washington also guarantees the country's protection under its nuclear umbrella.
Tensions on the Korean Peninsula are at their highest point in years after a series of missile tests by Pyongyang.
Hours before the test, North Korea's state news agency KCNA had released pictures showing Kim Jong Un inspecting a silver-coloured, hourglass-shaped warhead during a visit to the country's nuclear weapons institute, accompanied by scientists. The two countries technically remain at war since a truce agreed to end the 1950-1953 Korean War was never followed by a peace agreement.
The North didn't directly threaten the British authors of "North Korea Confidential: Private Markets, Fashion Trends, Prison Camps, Dissenters and Defectors", but said the book "viciously defamed and distorted" the country's realities. It says its weapons programmes are needed to counter USA aggression.