The gunman who shot dead six people at a Sikh temple in Wisconsin died of a self-inflicted gunshot wound to the head after he was shot by police, says the FBI.
The authorities said on Wednesday police responders shot gunman Wade Michael Page, 40, in the stomach, but they did not fire the fatal shot, as they had initially said.
Authorities previously said an officer had fired the shot that killed Page.
The FBI also confirmed Page's ex-girlfriend had been arrested on a weapons charge.
There is no suggestion Misty Cook, 31, was involved in Sunday's attack.
FBI special agent-in-charge Teresa Carlson told a news conference that investigators had not yet "clearly defined a motive'' for Sunday's shooting.
Authorities have not identified anyone other than Page as being responsible for the attack. Temple members have said the temple had never received any threats, and Page had not been seen at the temple in the past.
Carlson said federal officials had not opened any investigation into Page before the shooting. She said investigators were interviewing dozens of people who have known Page as they worked to determine for a possible motive.
On Wednesday, Barack Obama, the US president, called Manmohan Singh, India's prime minister, a Sikh, to express his condolences over the tragedy, the White House said.
The 40-year-old army veteran strode into the temple shortly before Sunday services and opened fire with a 9 mm pistol. The dead included temple President Satwant Singh Kaleka, who was shot as he tried to fend off the shooter with a butter knife.
Page wounded a responding police officer in the parking lot before another officer shot Page.
The FBI has classified the incident as domestic terrorism, a violent act for social or political gain.
Page had a record of minor alcohol-related crimes in Texas, Colorado and North Carolina. He was demoted during a stint in the army for getting drunk on duty and going absent without leave before he was discharged in 1998.
Page eventually moved to Wisconsin, living with a girlfriend and working at a factory.
The investigation could take weeks or longer.