Of that total, $8.6 bn is for training and equipment for Afghan security forces, and $2 bn for reconstruction.
Karzai called on the army, which split into factions during the 1992-1996 civil war, to commit to Afghan security.
"Our generals, officers and youth in the army should serve merely and only the national interests of the country and must not be at the service of any political ideology, political parties or factions," he said at the handover ceremony.
Before the collapse of communism in Afghanistan, its national army had a 500,000-strong force.
Equipped with Russian technology, it held more than 5,000 armoured vehicles and 400 planes.
By the time the ruling Taliban government was toppled in a US-led invasion in 2001, none of the army's technology was left.
Poorly equipped Afghan soldiers are currently fight alongside troops from Nato and US-led forces against Taliban fighters and its allies, including al-Qaeda.
The 40,000-strong Afghan army aims to reach 70,000 by 2009. However, defence officials have said a force three times the size is needed.
Abdul Rahim Wardak, Afghan defence minister, called on Nato for more help in building an independent and robust national army, saying it would "gradually make the army capable of defending our country in this violent part of Asia".
Also on Thursday, fighters killed two Pakistani government officials and a guard in an ambush near the Afghan border, according to a security official.
No one has claimed responsibility for the attack in the tribal region of North Waziristan.
The district-level officials were travelling in a car with a guard and three other people when gunmen ambushed them near the town of Mir Ali, a security official said.
Police returned fire but the attackers escaped, said the security official, who declined to be identified.
It is not known if any of the attackers were wounded or killed.