There is no timetable for the assembly to convene, and al-Jaafari and his alliance must agree with other elected parties on who will fill the three posts and the Cabinet. Even then, the prime minister has a month to name his Cabinet before the assembly vote.
Al-Jaafari's selection on Tuesday came after former Washington ally Ahmad Chalabi dropped out of the race following three days of round-the-clock bargaining. Al-Jaafari has been seen as having close ties to Iran's ruling clergy, though he denies any links to a government that President Bush has said is part of an "axis of evil."
For al-Jaafari, 58, to succeed, he'll have to meet conflicting demands from Kurds, Sunni Arabs and even Islamic hard-liners within his alliance
Iraq's secular Kurds and many Sunnis worry that al-Jaafari will try to impose his Dawa Party's brand of conservative Islam on the country, particularly because the assembly will be charged with writing a new constitution.
Al-Jaafari told the AP last week that Islam should be the official religion of Iraq "and one of the main sources for legislation, along with other sources that do not harm Muslim sensibilities."
He skirted his party's official position, which explicitly urges the "Islamization" of Iraqi society and the state, including the implementation of Shariah, or Islamic law.
"Theory is different from practice," al-Jaafari said.
Allawi also asked Iraq's minority Sunnis, who mostly boycotted the elections, to play a role in the new government. Such a move could help deflate the insurgency, mostly believed to be made up of Sunni Arabs that once belonged to Saddam's Baath party.
"The missions ahead of us are very big, above all is achieving national unity by action and not only by saying, and the integration of the Iraqi sectors which didn't participate in the elections," Allawi said.
Allawi has staunchly opposed de-Baathification - the effort to rid the government and administration of former Baath Party members.
A soldier from the U.S. Task Force Liberty was killed Wednesday when assailants set off the bomb near Tuz, 105 miles north of Baghdad, the military said.
At least 1,485 members of the U.S. military have died since the beginning of the Iraq war in March 2003, according to an Associated Press count.
The car bomb exploded in western Mosul, said Essam Youssef of the city's Jamhouri hospital. Its target was not immediately clear. Witnesses said no U.S. or Iraqi forces were in the area.
The U.S. military said two people were killed and 14 wounded in the attack.
Also in Mosul, U.S. soldiers shot and killed a civilian in a pickup truck who approached their convoy too closely to pass it, policeman Ahmed Rashid said. Weary of car bombs, most U.S. military vehicles carry signs warning drivers to keep away.
