In a statement sent to Al-Jazeera, the group, calling itself "Islamic Response," said it was holding Cpl. Wassef Ali Hassoun, a U.S. Marine of Lebanese heritage, but he was safe at a location they did not identify.
The United States reported Hassoun missing after he did not report for duty at his base in Iraq on June 20.
On June 27, Al-Jazeera broadcast a videotape showing Hassoun blindfolded, along with a statement from militants threatening to kill him unless the United States released all Iraqis in "occupation jails." Militants held a curved sword over his head.
Hassoun's fate was thrown into confusion Saturday when a group identifying itself as the Ansar al-Sunna Army, another militant group, posted a note on an Islamic Web site saying it had killed him.
The group posted a denial of the killing Sunday on its own Web site.
"The denial gave us a big relief," Hassoun's brother, Sami, told The Associated Press by telephone from the northern city of Tripoli, where some of Hassoun's relatives live.
Hassoun's family had been hoping his Arab roots and religion would give the kidnappers reason to show mercy, reports CBS News Correspondent Lee Cowan. But out of all the hostages taken by Islamic militants, he is only Muslim hostage who hasn't been released.
The group who claimed to have taken that video more than two weeks ago tells the al-Jazeera network that Hassoun is still alive, in part because he has promised not to return to the U.S. military, reports Cowan.
Other militant groups have captured and threatened to behead other foreign Muslim hostages, creating an uproar among many Muslims, including other militants. All the captured Muslims aside from Hassoun have been released unharmed.
The statement Monday claimed that Hassoun had promised not to return to the American military.
The statement was issued in the name of the same group that claimed initial responsibility on the June 27 video for the kidnapping. The group calls itself "Islamic Response," the security wing of the "National Islamic Resistance 1920 Revolution Brigades." The name refers to the uprising against the British after World War I.
Tarek Nosseir, a spokesman for the Hassoun family in West Jordan, Utah, declined comment Sunday.
Family members said Hassoun was born in Lebanon, educated at American schools there and then joined the Marines after moving to the Salt Lake City area. Hassoun, fluent in Arabic, French and English, was serving in the Marines as a translator in his second stint in Iraq when he was captured.
Hassoun's father, Ali Hassoun, who lives in Tripoli, Lebanon, repeatedly pleaded for his son's release. He and his other sons have contacted politicians and Muslim clerics in Lebanon and Islamist groups in Iraq in hopes of securing the Marine's release.
The Iraqi guerrilla group denied on Sunday that it put out a statement a day earlier claiming to have beheaded kidnapped Cpl. Wassef Ali Hassoun, leaving the fate of the U.S. Marine unclear.
The denial by the Ansar al-Sunna Army group left open the possibility that the 24-year-old American of Lebanese origin was killed by another group or that he was still alive.
A Lebanese Foreign Ministry official in Beirut said Hassoun was believed to be dead and that Lebanese officials in Iraq were trying to track down his body.
The denial from the Ansar al-Sunna Army came a day after a statement in the group's name announced that Hassoun had been beheaded. The Ansar al-Sunna said Sunday they didn't issue the statement, leaving it unclear if the 24-year-old was killed by another group or was still alive.
But with conflicting reports and no hard evidence, the family had remained afraid for Wassef's life and was still reeling from the possibility he had been beheaded.
"We are hoping that good news will come... that Wassef is alive, God willing," Sami Hassoun said. He renewed his appeal to the kidnappers to release his brother.
In other developments:
The report that Wassef Hassoun had been killed came Saturday in a message posted on Islamic radical Web sites, signed by the Ansar al-Sunna Army in Qaim, a hotbed of guerrilla activity on Iraq's border with Syria. That name was different from the one given in the statement that originally announced Hassoun's abduction a week ago.
The Lebanese Foreign Ministry announced Sunday that it had independent information from Baghdad that he had been killed.
But after strongly condemning the death Sunday, Foreign Minister Jean Obeid later said news of the death "was not official."
Obeid said the Lebanese charge d'affaires in Baghdad was "in contact with some forces that have indirect links to the (kidnappers), and these forces say they lost hope in all attempts (to win his release) and by last night the group was about to behead him or had already beheaded him."
The U.S. military in Baghdad said it was checking into the claim of Hassoun's death but had no confirmation.
In its statement Sunday on its official Web site, the Ansar al-Sunna Army, which has taken responsibility for suicide bombings and other attacks in the past, said it had nothing to do with the claim of Hassoun's slaying the day before.
"In order to maintain our credibility in all issues we declare that this statement that was attributed to us has no basis of truth," it said.
It added that "any statement that is not issued through our site doesn't represent us."
In West Jordan, Utah, where Hassoun lived with his eldest brother Mohammed after moving to the United States in the early '90s, relatives were in seclusion since the posting of the death report Saturday.
A telephone message left early Sunday morning at the home of Mohammed Hassoun was not immediately returned.
On Saturday, Shuaib-Ud Din, the imam at Khadeeja mosque in nearby West Valley City, met with Hassoun's family members for about 15 minutes at their home, where the yard had been decorated in recent days by about two dozen flags put up by Boy Scouts.
At a news conference at the mosque, the imam said the Hassouns were praying and awaiting official word of Wassef's fate. He cautioned the public against automatically believing reports out of the Middle East.
"Every family has a different way of dealing with the crisis. This family prefers less attention," Shuaib said. "They don't like the media outlets to be pounding on their door. They would like some privacy."
Hassoun, fluent in Arabic, French and English, was serving the Marines as a translator in his second stint in Iraq when he was captured.
The original claim of Hassoun's abduction was issued in the name of "Islamic Response," the security wing of the "National Islamic Resistance - 1920 Revolution Brigades," rather than the Ansar al-Sunna Army.
On June 27, the Arab television station Al-Jazeera broadcast a videotape showing Hassoun blindfolded, along with a statement from militants threatening to kill him unless the United States releases all Iraqis in "occupation jails."
Since Hassoun's capture, his father, Ali Hassoun, who lives in Tripoli, repeatedly has pleaded for his son's release, saying he was not involved in the fight against Iraqi resistance groups. He and his other sons had contacted politicians and Muslim clerics in Lebanon and Islamist groups in Iraq in hopes of securing the Marine's release.
Family members said Wassef was born in Lebanon, educated at American schools there and then joined the Marines after moving to the Salt Lake City, Utah, area.
Arabs working with the Americans in Iraq also have been targets of Iraqi insurgents, and at times the motives for kidnappings have not been clear.
Militants have kidnapped at least five Lebanese hostages in Iraq in recent months -- all for financial reasons. Four of those kidnapped were later released but one, Hussein Alyan, was shot dead and his body dumped beside a road.