The government said in a statement, quoted by state television on Monday, that Garang, six of his companions and a crew of seven had died after the helicopter left Uganda's capital on Saturday evening.
"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," the statement said.
"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."
The statement added: "The march of peace will continue towards its goal and his death will only make us stronger and more determined to complete the march which he began and his companions began."
The bodies of the dead were found early on Monday in rugged mountains running along the Sudanese-Ugandan border.
Ugandan President Yoweri Museveni said he would appoint a special panel to investigate the crash.
In addition, Museveni said Uganda had asked a foreign government to look into Saturday's crash to definitively establish that it was an accident, as officials in Kampala have insisted, and not the result of sabotage or terrorism.
Garang's Sudan People's Liberation Movement (SPLM) confirmed his death and vowed to continue his work.
"Sudan has lost its loved son Dr John Garang," SPLM deputy leader Salva Kiir Mayardit told a news conference in Nairobi, where SPLM members wept in grief.
President Omar al-Bashir's office
says peace efforts will continue
"We in the SPLA (Sudan People's Liberation Army) and SPLM will continue his vision for a peaceful Sudan."
The Sudanese military said the aircraft had left the Ugandan capital of Kampala for southern Sudan on Saturday at about 6.30pm.
Contact was lost about an hour later, Sudanese Information Minister Abd al-Basit Sabdarat said on Sunday.
Ugandan, Sudanese and Kenyan troops have been hunting for the helicopter since early Sunday, after it failed to reach its destination in southern Sudan on Saturday.
The helicopter had tried to land in the New Kush region of southern Sudan, but aborted the landing because of bad weather and headed back south, Ugandan President Yoweri Museveni said in a statement early on Monday.
Garang's aircraft was last heard
near the Kenyan and Sudan border
Garang had left Uganda after talks with Museveni.
The helicopter was last heard near Pirre, a mountainous region near the Kenyan and Sudanese borders on the edge of a large national park, he said.
Uganda said earlier on Monday that it had been searching for Garang since Sunday morning in the Kidepo area, which lies on the border with Sudan.
"Since [Sunday] morning, we have been searching the Kidepo area to locate the chopper without success," a statement issued by Museveni said.
There had been conflicting accounts of Garang's fate and the aircraft. Sudanese officials said it was a plane and Ugandan officials claimed it was a Ugandan government helicopter.
Garang waged a two-decade-long war against Khartoum from southern Sudan until a historic peace deal in January, under which he became first vice-president on 9 July.
He played a key role in reaching that peace deal earlier this year, ending a civil war that had killed more than two million people.
Garang, who returned to Khartoum in early July for the first time since the 1983 launch of the war, also took the oath as head of a new autonomous administration for south Sudan.
His swearing in followed the promulgation of a new power-sharing constitution provided for under January's peace agreement.
Garang (L) was sworn in as first
vice-president on 9 July
The head of the African studies centre in Khartoum, Dr Hasan Makki, told Aljazeera that Garang would be irreplaceable.
"The consequences after Garang's death will be great as Garang represented many symbols," Makki said.
Makki added that he was a key figure in countering separatist calls within the SPLM.
"There are fears that calls of separation might rise as Garang was the only one who could have curbed separatists inside the SPLM," he concluded.
Thousands gathered in southern Sudan's provisional capital of Rumbaik on Monday to mourn the death of Garang, who many there regard as the region's main hope for peace, an official said.
Residents of the town, the provisional capital of an autonomous south under a landmark January peace deal Garang signed with Khartoum, turned out in large numbers at Rumbaik's Freedom Square to remember the man who just three weeks ago was sworn in as Sudan's first vice-president, the official said.
Many gathered in southern Sudan
to mourn the death of Garang
The situation was sombre but calm, and there were no outbursts of violence like the rioting reported in Khartoum, according to a former commander of Garang's SPLM army, now a political operative in Rumbaik.
"The town of Rumbaik has been informed of this national disaster," Pagan Amum told AFP in Nairobi by satellite phone from the dusty outpost about 900km south of Khartoum.
"The situation is calm," he said. "People have taken the sad news with courage and are ready to forge ahead despite the sad news."
Under the terms of the 9 January peace agreement, Rumbaik is to serve as the temporary capital of autonomous southern Sudan before the government moves to the larger town of Juba.