"Today, at around 12:36pm, we detected an artificial natural disaster measuring 5.6 magnitude... we are analyzing whether a nuclear test was conducted", South Korea's Joint Chiefs of Staff said, according to Yonhap.
China has expressed annoyance at North Korea's frequent ballistic missile launches, but analysts have said China probably would not take serious action - unless there is another nuclear test.
The Earthquake Administration said in a statement on its website that the quake, which occurred around 3:30 GMT, was recorded at a depth of zero kilometres.
North Korea has conducted five nuclear missile tests since 2006, including two previous year.
Questions remain over whether Pyongyang has successfully miniaturised its weapons, and whether it has a working H-bomb, but KCNA quoted Kim as saying that "all components of the H-bomb were 100 percent domestically made".
"It follows two such tests last year and more than a dozen missile tests this year".
After the ICBM tests, the United Nations imposed tough new sanctions on North Korea that included a ban coal and other exports worth over $1 billion.
"Today's test is a highly risky affront to the entire global community". Analysts say Kim wants nuclear power-to nuclear power conversation with the USA without the South's involvement.
The agency reported the new H-bomb could be fitted and would be loaded into an intercontinental ballistic missile, which experts believe would be able to reach the US.
Earlier, Japan confirmed its near neighbour had conducted a sixth nuclear test.
The U.S. Geological Survey called the first quake an explosion with a magnitude 6.3.
North Korea's latest claims have not been independently verified.
The report by North Korea's official KCNA news agency comes amid heightened regional tension following Pyongyang's two tests of ICBM missiles in July that potentially could fly about 10,000 km (6,200 miles), putting many parts of the mainland United States within range.
Images of Kim Jong-un purportedly signing the suspected nuclear test ran on North Korean state TV.
The statement was enough to raise concerns in Washington, Japan and the Korean Peninsula about the state of North Korea's nuclear arsenal.