Chang Song-thaek was dramatically removed from a Communist Party session by armed guards earlier this week.
It was the biggest upheaval since Mr Kim succeeded his father two years ago.
State news agency KCNA said Mr Chang had admitted at a military trial on Thursday to attempting to overthrow the state, and was executed immediately.
Mr 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 harbouring his own political ambitions, it said in a lengthy and detailed report.
In the US, the White House said it could not independently verify the reports but had "no reason to doubt" them.
"If confirmed, this is another example of the extreme brutality of the North Korean regime. We are following developments in North Korea closely and consulting with our allies and partners in the region," it said in a statement.
An image of Mr Chang in court, his hands bound, was released on Friday
Before his fall from grace, he was frequently pictured beside leader Kim Jong-un
South Korean President Park Geun-hye said the North was carrying out a "reign of terror"
Mr Chang - married to the elder Kim's sister - had held senior posts in the ruling party and the National Defence 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 Mr 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 colours as 'reformist' known to the outside world would help his 'new government' get 'recognised' by foreign countries in a short span of time."
It also said Mr Chang:
Attempted to "overthrow the state"
Transformed his department into "a 'little kingdom'" and attempted to "trigger off discontent" within the army to mobilise a coup
Took control of the "major economic fields of the country" and "schemed to drive the economy of the country and people's living into an uncontrollable catastrophe"
Committed corruption by transferring construction units to his contacts
Committed irregularities related to a joint economic zone with China, Rason
Was responsible for unpopular currency reforms in 2009. In December 2009 Pyongyang's reported redenomination of the won knocked two zeros off the nominal value of each banknote.
Mr Chang admitted his "crimes" in court and a death sentence was "immediately executed", KCNA said.
Economic reforms?
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, Mr 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.
The BBC's Lucy Williamson in Seoul says there are various theories surrounding Mr Chang's demise, including suggestions that his work with China had led him to admire some of Beijing's economic reforms.
But it is more likely that he presented a perceived threat to his nephew's authority, says our correspondent.
As news of the purge emerged earlier this week, South Korean President Park Geun-hye warned the North was "carrying out a reign of terror" to reinforce Mr Kim's position.
She said the volatile relationship between the two countries was likely to become "more unstable" as a result.
On Friday, South Korea's military said it had tightened surveillance on Pyongyang, news agency Yonhap reported.
Unification Ministry spokesman Kim Eyi-do said the government had "deep concerns" about the latest developments and was "watching the situation closely".
Meanwhile, Japan's top government spokesman Yoshihide Suga said: "We will calmly monitor the situation while communicating with other countries and collect relevant information."
Chinese state media reported on Mr Chang's execution, but there has been no official government response so far.
Source:BBC
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