In an official statement in early afternoon, the Cabinet named the crown prince, Shaikh Saad al-Abdulla al-Sabah, the 76-year-old crown prince, as his successor.
But because illness has incapacitated Saad, political analysts expect Shaikh Sabah al-Ahmad al-Sabah, the prime minister, to run the country - a role he has played over the past four years.
The illness in the past few years of Jaber and Saad had sparked concerns at home and abroad over who would rule the Opec nation, which has on one-tenth of the world's oil reserves and was the main launch pad for the 2003 US-led invasion of Iraq.
"With the utmost of sorrow and sadness, the (royal court) announces to the Kuwaiti people, the Arab and Islamic nations and the peoples of friendly world nations the death of His Highness Sheikh Jaber al-Ahmad al-Jaber al-Sabah," said a royal court statement carried by the state news agency KUNA.
"With the utmost of sorrow and sadness, the (royal court) announces to the Kuwaiti people, the Arab and Islamic nations and the peoples of friendly world nations the death of His Highness Sheikh Jaber al-Ahmad al-Jaber al-Sabah"
A royal court statement
It said the amir, who had ruled Kuwait since 31 December 1977, "passed away at dawn on Sunday". Jaber had been ailing since suffering a brain haemorrhage in 2001. He had surgery on his leg in the United States in May.
Former Iraqi President Saddam Hussein's troops occupied Kuwait for seven months before US-led forces expelled them in the 1991 Gulf War.
The amir returned from exile in Saudi Arabia to lead the rebuilding of a land scarred by killings, torture and brutal rule during the occupation. He also oversaw the rehabilitation of oilfields set on fire by retreating Iraqi troops.
Kuwait said there would be a 40-day official period of mourning and that government offices would be closed for three days from Sunday.
The amir was the 13th ruler of a 245-year-old dynasty that has ruled Kuwait since the Anaiza tribe, to which the al-Sabahs belonged, migrated from the Arabian hinterland.
Shaikh Sabah al-Ahmad al-Sabah,
the prime minister, runs the country
Since the fall of Saddam in neighbouring Iraq in 2003 and US calls for change in the Middle East, the ruling family had come under intense pressure from both Islamists and pro-Western liberals to loosen its grip on the government and share power.
The ruling family has also been under pressure from parliament and elders within its ranks to break with tradition and replace the ailing crown prince.
The succession process alternates between the two branches of the ruling family, al-Jabers and al-Salems.
Strong US ally
Kuwait, a founder Opec member, enjoys one of the world's highest standards of living, despite its reliance on oil exports, unpredictable oil income and huge losses from the 1990-1991 Iraq occupation.
Kuwait hosts up to 30,000 US troops and about 13,000 US citizens live in the country.
Kuwait hosts up to 30,000 US
troops in the country
Kuwait has cracked down on Islamists opposing the US military presence in the country.
Diplomats say radical Islam is taking hold among Kuwaiti youth.
In December, a Kuwaiti court sentenced to death six suspected militants linked to Osama bin Laden's al-Qaida for bloody attacks in the country.
The six were among 37 Islamists on trial as members of the Peninsula Lions group, believed to be linked to al-Qaida in Iraq and Saudi Arabia.