Kabul, Afghanistan (CNN) -- An elite U.S. special forces team rescued an American doctor who had been abducted in Afghanistan, but lost one of their own members in the mission, officials said.
Dr. Dilip Joseph was freed 11 hours after his captors released two other kidnapped staffers of his nonprofit agency, Morning Star Development, the organization said Sunday.
Hours later, U.S. Defense Secretary Leon Panetta announced that "a U.S. service member was killed in the operation."
A U.S. official said the man who was shot dead belonged to the Navy's Special Warfare Development Group, more commonly known as SEAL Team Six. The elite unit is the same one that took part in the raid that killed al Qaeda leader Osama bin Laden, but the official didn't know if the fallen service member was involved in that operation.
While he did not delve into detail, Panetta said, "the special operators who conducted this raid knew they were putting their lives on the line to free a fellow American from the enemy's grip."
The NATO-led International Security Assistance Force said Joseph was believed to be in imminent danger of injury or death when the rescue mission was launched.
"He gave his life for his fellow Americans, and he and his teammates remind us once more of the selfless service that allows our nation to stay strong, safe and free," President Barack Obama said of the slain U.S. service member, who was not immediately identified.
There were conflicting reports about who was behind the kidnappings. ISAF said it was Taliban insurgents, while two local Afghan officials told CNN it was smugglers.
The kidnapping of Joseph and the two other Morning Star staff members occurred around 3:30 p.m. Wednesday (6 a.m. ET), as the three men were returning from a rural medical clinic in eastern Kabul province.
"They were stopped and captured while driving, by a group of armed men," according to the agency, which sponsors community and economic initiatives in Afghanistan.
"They were eventually taken to a mountainous area about 50 miles from the Pakistan border."
Tribal leader Malik Samad and district chief Muhammad Haqbeen told CNN that Joseph and an Afghan doctor were abducted near the village of Jegdalek in the Sarobi district, just outside Kabul.
Morning Star said negotiations began "almost immediately" between the captors, the hostages and the agency's crisis management team in Kabul and Colorado Springs, Colorado. In addition to U.S. and Afghan authorities, the agency thanked "Afghan residents (including) elders and local leaders who made visits and appeals to the captors advocating for the release of the hostages."
The "on-again, off-again" negotiations lasted three days into Saturday night, when two of the three were released.
Morning Star did not release the identities of the two men, citing safety concerns "because they live and work in the general region." The group said one of them belonged to its medical staff and the other was part of the organization's support staff.
The two men got out of the area and eventually were taken to a police station, at which time Morning Star and their families learned they'd been released.
The Afghan doctor's family paid $12,000 to the smugglers, who later released him, Haqbeen and Samad said. For its part, Morning Star stated "categorically that we paid no ransom, money or other consideration" to anyone in securing any of its staffers' release.
Sometime Saturday night, military forces rescued Joseph in a mission ordered by Gen. John Allen, commander of U.S. forces in Afghanistan. Samad and Haqbeen, the two Afghan officials, said the operation took place in Laghman province.
"Today's mission exemplifies our unwavering commitment to defeating the Taliban," Allen said in a statement. "I'm proud of the American and Afghan forces that planned, rehearsed and successfully conducted this operation. Thanks to them, Dr. Joseph will soon be rejoining his family and loved ones."
Joseph has worked with Morning Star for three years. He serves as its medical adviser, and travels frequently to Afghanistan, the agency said.
The doctor appeared to be in "good condition and uninjured" at Bagram Airfield after his abduction. Morning Star said "he will receive precautionary examinations and debriefing before returning to his Colorado Springs home, probably within a few days."
His family issued a statement Sunday acknowledging "harrowing" uncertainty in recent days and extending its "deepest condolences to the family of the American sailor who died during Dilip's rescue."
"We are incredibly grateful for the multiple agencies of the U.S. government that have supported us in this difficult time, and especially the quick response by our military and partner allies to rescue Dilip," the family said. "They showed great heroism and professionalism."
Despite the kidnappings of Joseph and its two other staffers, Morning Star reiterated its "commitment to continue its work" in Afghanistan.
CNN's Barbara Starr, Maria Ebrahimji, Deborah Doft and Greg Botelho contributed to this report.