The brazen, daytime attack was against Aquila al-Hashimi, one of three women on the council, a Shiite Muslim and a strong candidate to become Iraq's representative at the United Nations. Al-Hashimi had been preparing to leave for a key U.N. General Assembly meeting in New York on Tuesday.
Al-Hashimi, a Shiite, is the only official from the regime of ousted President Saddam Hussein to have been appointed to the 25-member council. She was a career diplomat in Saddam's Foreign Ministry, participating in international tours to promote the regime's policies.
The Governing Council president blamed Saddam Hussein loyalists for the shooting. U.S.-led forces have been struggling to put down a guerrilla-style insurgency that has targeted Americans and Iraqis who work with them and has hampered reconstruction.
CBS News Correspondent Elizabeth Palmer says the attack is "being interpreted as another brutal warning to all those who cooperate with Americans in the stabilization of Iraq."
The attack came at 9 a.m. when gunmen fired a rocket-propelled grenade on al-Hashimi's car soon after she left her house in western Baghdad, members of her security detail said. The grenade missed, and the attackers opened fire with assault rifles.
Al-Hashimi, critically wounded in the abdomen, was rushed to the al-Yarmouk Hospital for surgery and was later moved in a convoy of American armored vehicles and military ambulances to the U.S. military hospital at Baghdad International Airport. Three of her bodyguards were also wounded.
There, she was reported in stable condition. "She is fine," said Haitham al-Husseini, an adviser to council member Abdel-Aziz al-Hakim, a fellow Shiite.
In other developments:
Ahmad Chalabi, the president of the Governing Council for September, said al-Hashimi's attackers "were remnants of the Baathist regime and Saddam's assassins," referring to Saddam's former ruling Baath party.
"The members of the Governing Council and ministers will not be intimidated by the terrorists," Chalabi said in a statement. "They will continue to do their patriotic duty to move Iraq towards freedom, democracy and sovereignty."
Baghdad police commander Brig. Gen. Ahmed Ibrahim told The Associated Press that no one had been arrested in the attack and he refused to say who might be behind it.
The Governing Council was established by the U.S.-led coalition in mid-July to put an Iraqi face on the process of rebuilding the country. That process has been hampered by unflagging violence blamed on Saddam loyalists, including ambushes of American troops, attacks on Iraqi police and others helping U.S. forces and sabotage of infrastructure.
The White House denounced the assassination attempt. Spokeswoman Suzy DeFrancis called it a "tragic situation" that is a part of a "continuing pattern" in which insurgent forces attack signs of success of an Iraqi transition to democracy.
L. Paul Bremer, the U.S. civilian administrator for Iraq, called the shooting "an attack against the people of Iraq and against the common goals we share for the establishment of a fully democratic government."
Al-Hashimi has emerged as a leading foreign policy figure on the council, participating in a delegation that addressed the United Nations in July. At Tuesday's General Assembly session, the council delegation will try to assume Iraq's U.N. seat — and if it succeeds, many U.N. diplomats expected al-Hashimi to be named Iraq's representative.
Chalabi said in his statement that the council delegation would attend the U.N. session, but did not say whether al-Hashimi would be replaced.
Unlike heavyweight members of the Governing Council, such as Chalabi, lesser known members are not heavily protected. Chalab is surrounded by men from his Free Iraqi Forces militia, but al-Hashimi relies on her brother in a country where prominent people often use relatives, clan members or men from their hometowns because they trust them.
Middle East analyst Hazhir Teimourian tells CBS News Reporter Charles D'Agata many governing council members are particularly vulnerable because they avoid surrounding themselves with US troops, in an effort to appear more accessible to the Iraqi people.
Last month, the leader of a Shiite movement represented on the Governing Council, Ayatollah Mohammed Baqir al-Hakim, was assassinated in a bomb blast in the holy city of Najaf south of Baghdad. The attack, widely thought to be the work of Saddam's supporters, killed at least 85 people. Al-Hakim's brother, council member Abdel-Aziz, now heads the Supreme Council of the Islamic Revolution in Iraq.
The continuing security crisis has raised questions about America's stewardship of Iraq since President Bush declared an end to major combat operations May 1. Since then, 82 American soldiers and 11 Britons have been killed in hostile encounters.
U.S. troops have been trying to track down pro-Saddam fighters who have launched near-daily attacks on U.S. troops, including an ambush and gunbattle that killed three soldiers and wounded two on Thursday night near Tikrit.
U.S. tanks and armored fighting vehicles rumbled through Tikrit early Saturday in a show of force meant to discourage more attacks and flush out armed resistance.
Fifty-eight Iraqis were captured after the attacks on Thursday, described as some of the fiercest and best-planned resistance in months. U.S. troops seized a considerable number of weapons from a minivan fleeing the area, the military said.
During Saturday's patrol, tanks swept through residential areas, occasionally dismounting to set up security points, to check cars and people leaving Tikrit after the city's 11 p.m. curfew.
The patrol ended without incident.