KCNA news agency said the American, named "Miller Matthew Todd", 24, had been taken into custody on 10 April.
This was due to "his rash behaviour in the course of going through formalities for entry" into North Korea, it added.
The news was released as US President Barack Obama held talks with his South Korean counterpart on his Asia tour.
KCNA said Mr Todd had torn up his tourist visa, shouting that he had "come to the DPRK (Democratic People's Republic of Korea) after choosing it as a shelter".
The agency said such an action constituted a "gross violation" of North Korean law.
No reason was given for the two-week delay in the announcement of his detention.
The US has no ties with Pyongyang, with Sweden usually acting on its behalf in cases involving US citizens.
The US state department said it was aware of the report and had been in touch with the Swedish embassy.
North Korea is currently holding US-Korean missionary Kenneth Bae, who was arrested in November 2012.
He is serving 15 years of hard labour after being convicted of trying to overthrow the government.
Efforts from Washington to secure Mr Bae's release have so far been unsuccessful.
Merrill Newman, an 85-year-old US national, was briefly held by North Korea last year.
He was freed after confessing to committing crimes during the Korean War - a statement he said was given under duress.
US President Obama earlier said America stood "shoulder to shoulder" with South Korea over North Korean provocation.
He spoke after holding talks with South Korean leader Park Geun-hye.
North Korea has detained a 24-year-old US tourist, reportedly for "rash behaviour" at immigration, the state news agency says.