Chang Song-thaek was dramatically removed from a Communist Party session by armed guards earlier this week.
It was the biggest upheaval since Kim succeeded his father two years ago.
State news agency KCNA said Chang had admitted at a military trial on Thursday to attempting to overthrow the state, and was executed immediately.
Chang, who is thought to have mentored his nephew during the leadership transition from Kim Jong-Il to his son Kim Jong-Un in 2011, was "worse than a dog", said the agency.
He had admitted abusing his positions of responsibility to form a faction against the state and to harboring his own political ambitions, it said in a lengthy and detailed report.
Chang - married to the elder Kim's sister - had held senior posts in the ruling party and the National Defense Commission, the North's top military body.
He was frequently pictured alongside his nephew and seen by some observers as the power behind the throne.
But in early December, it emerged that he had been removed from his senior military position and that two of his aides had been executed.
Then on Monday, KCNA broadcast footage of him being removed from a party session by uniformed guards.
In a long report on Friday, KCNA described Chang as a "traitor" and "human scum".
It said: "Chang dreamed such a foolish dream that once he seizes power by a base method, his despicable true colors as 'reformist' known to the outside world would help his 'new government' get 'recognized' by foreign countries in a short span of time."
Analysts say his fall from grace could be seen as the latest in a series of carefully calibrated moves to demonstrate Kim Jong-Un's authority and an assertion of his independence.
In August 2012, Chang made a high-profile trip to China, where he met then-President Hu Jintao. The two sides later signed a raft of economic deals, including the development of two special economic zones: Rason, on North Korea's east coast, and Hwanggumphyong, on the border with China.
BA/BA