Mr. Kim, who had been teaching accounting in North Korea, is the third US national now being detained in North Korea.
North Korea, which is understood to have large prison camps where it holds dissidents and a handful of foreign prisoners, has in the past been accused of using detained Americans as "human shields."
Mr. Kim, 58, had been working at the Pyongyang University of Science and Technology for several weeks.
His arrest was reported last month but only now are details emerging.
The university, which opened in 2010, was founded by evangelical Christians and teaches students from the country's elite.
But all foreigners are treated with suspicion in North Korea and the US government generally advises American nationals not to go there.
It could be that the detention of the three American nationals is an attempt by Kim Jong-un to seek a way of opening secret back channel communications with the US.
In 2010, former President Jimmy Carter was encouraged to come to North Korea, ostensibly to negotiate the release of an American, Ajjalon Mahil Gomes, with Kim Jong-il.
The year before that, former President Bill Clinton came to Pyongyang to obtain the freedom of journalists Laura Ling and Euna Less.
Carter also visited North Korea in 1994 — and met Kim Il-sung — and on all three occasions the former presidents were used as back channels.
University of Georgia professor Han Park, who helped arrange both of Carter's trips, said years later: "The release was not the North Korean purpose. They wanted to have a much more substantive discussion when the official line is seemingly blocked."
It is possible Kim Jong-un may be using the detained Americans to open the way for another former President — possibly George W. Bush — to visit North Korea and open a back channel to Trump in a bid to avert conflict.
Two other US citizens — Otto Warmbier and Kim Dong Chul — are serving prison sentences in North Korea.
@realDonaldTrump Amercians held in North Korea — Otto Warmbier and Kim Dong-chul. They are serving 15 year sentence with hard labor— Kalpesh Patel (@kpatel63) February 4, 2017
Kim Dong Chul, a pastor, was sentenced to 10 years of hard labor last year for spying. In 2016, he was allowed to do an interview with CNN in which he admitted to espionage.
Warmbier was sentenced to 15 years for stealing a propaganda poster and for "crimes against the state."
The Democratic People's Republic of Korea (DPRK) considers itself virtually at war with South Korea and the United States and tensions have risen in recent months since President Trump took over in the White House.
The Korean Central News Agency said Kim Sang Dok "was intercepted for committing criminal acts of hostility aimed to overturn the DPRK not only in the past but also during his last stay before interception."
Pyongyang has conducted numerous nuclear and ballistic missile tests in defiance of international bans and UN sanctions.
But Kim Jong-un has expressed outrage at massive joint military exercises conducted by the US and South Korea and by the introduction of the THAAD missile system.
On April 30, Trump said he would be "honored" to meet North Korea's leader "under the right conditions" and praised him as a "smart cookie" for seizing power after the death of his father, Kim Jong-il in 2011.