Sudanese President Omar al-Beshir confirmed Dr Garang’s death, adding that he remains committed to Sudan’s peace process.
Earlier, Ugandan officials said all passengers and crew on board the Ugandan helicopter died in the crash, near the Kenyan border.
The chopper has been missing since Saturday.
Dr Garang, the head of the Sudan’s People Liberation Movement, became vice president only three weeks ago after a peace deal was struck ending 21 years of civil war in the country.
“The presidency has followed the reports about the disappearance of the aircraft of Sudanese Vice-President John Garang and it is confirmed beyond doubt that it crashed in the Amatonj mountains,” said a statement read by President al-Beshir on national television.
“It resulted in the death of John Garang and six people accompanying him as well as seven members of the crew of the Ugandan presidential aircraft,” he said.
On January 9, Dr Garang and Vice President Ali Osman Taha signed the landmark deal ending the war between Sudan’s north and south, Africa’s longest-running conflict.
“We guarantee that the peace process will continue progressing in the same direction,” said the President al-Beshir.
“His passing will only reinforce our determination to pursue the peace process he and his comrades had started with the Sudan People’s Liberation Movement,” he said.
“By announcing this terrible event, the president of the republic extends his condolences to the entire Sudanese people and the world for the death of a man who believed in peace and worked towards it with sincerity, determination.”
A Ugandan official said it is believed bad weather was the cause of the crash.
Search crews reportedly reached the site on Monday and found the body believed to be that of Dr Garang’s, according to a United Nations official.
Dr Garang had been in Uganda meeting with Ugandan President Yoweri Museveni, as well as EU and US diplomats.
The US-educated economist-turned-guerrilla steered the SPLA through the bloody war over two decades against the government in the north, ruling with an iron fist.
He was also installed as the head of Sudan’s new southern autonomous region following the power-sharing constitution that was provided for under January’s peace agreement.
Dr Garang was considered to be an adept politician who allied himself with communists, courted US Christian groups and juggled tribal rivalries to retain power.