ALBUQUERQUE, N.M. — A faint sound made Albert and Peter Kottke stop and look around as they hiked out of the Gila Wilderness at the end of backpacking trip. A figure moved on the other side of the Gila River. As it drew closer, the two university students saw a woman, hunched over and moving slowly.

The Kottkes crossed the river to find Carolyn Dorn, 52, who had been alone in the Gila National Forest for five weeks after becoming trapped on the wrong side of the rain- and snow-swollen river. The search for her had been called off two weeks ago.

The brothers said they realized Ms. Dorn was too weak to go with them. They gave her food — Tang, almonds, dried apples, an energy bar, some hot soup, and a little cheese — scavenged firewood for her from the other side of the river, filled her water bottles and left her a book — suspense author Michael Connelly’s “Chasing the Dime.”

They hiked 20 miles over the next day and a half, and on Saturday hitchhiked into Silver City, where they contacted authorities.

“We got her prepared to spend another couple of nights while we went upstream to get help,” Albert Kottke, 25, a doctoral student in civil engineering at the University of Texas at Austin, said Sunday from his parents’ home in St. Paul, Minn.

A New Mexico National Guard helicopter crew, using night vision goggles and a U.S. Geological Survey map that the Kottke brothers marked, rescued the weak and dehydrated South Carolina woman before dawn Sunday and flew her to Silver City.

“It is a miracle she came out alive,” said search and rescue coordinator Frankie Benoist.

Ms. Dorn was about six miles from the nearest road in an area where the brothers, who have hiked in the region several times over the past two years, had never seen another human being, said Peter Kottke, whose 20th birthday was Sunday.

When they left, she was “very alert, talkative,” said Peter Kottke, a junior geological engineering major at the University of Wisconsin at Madison. “She seemed very relieved that somebody had finally found her.”

Ms. Dorn told the brothers she was warm enough at night, but her eyes lit up when they offered her the book, he said. He felt comfortable leaving her after that because “you could tell she had a positive outlook,” he said.

Temperatures in the area have dropped into the low teens overnight in recent weeks, according to the National Weather Service.