WASHINGTON (AFP) - A French couple died while hiking in the New Mexico desert but authorities rescued their nine-year-old son after finding photographs of him on his dead mother's camera, officials said.

David Steiner, 42, and Ornella Steiner, 51, both from the small town of Bourgogne near Reims, had been hiking in the White Sands National Monument on Tuesday afternoon when they apparently died from heat exhaustion, Otero County Sheriff's Detective David Hunter said.

"They underestimated the elements," Hunter told AFP on Friday, adding that autopsy results were still pending.

Park rangers found the mother first while on a routine patrol. They checked her camera and saw that she had been with two other people, Hunter said.

Authorities followed her tracks and found the body of the father, with the son nearby. He was dehydrated but responsive and was taken to a local hospital for treatment.

The New York Daily News said authorities had contacted the French consulate and the boy's grandmother had flown from France and taken custody of him on Thursday.

Otero County Sheriff Benny House told local media that the family had been traveling across the United States, visiting national parks and monuments.

The officers who rescued the boy found two empty 0.6-litre bottles with them. The boy told the officers they had been full of water when they set out at White Sands.

"The father and mother would take one drink while they made the child take two swallows of water," House told the Alamogordo Daily News.

"It might have been why the child fared so well due to his smaller stature, plus he probably consumed more water than they did," House was quoted as saying.

House said the boy told authorities that his mother had headed back toward the car after becoming ill on the hike. He carried on with his father.

House said temperatures had been around 38 deg C, according to the Alamogordo Daily News.

National Parks spokesman Patrick O'Driscoll told the newspaper there was no shade or water along any of the trails in the dune-filled White Sands National Monument.

Prior to this, two people this century have died of exposure while hiking in the park, in 2009 and 2011, the Alamogordo paper said.

Ornella Steiner had worked for about 20 years for the mayor's office in Reims.

"She was a very involved person, very vibrant, active and full of life," said deputy mayor Kim Duntze, who worked with her.

The Steiner couple loved traveling and had visited the southern United States for 10 days in 2014.

"When they came back from that trip, they often said: 'We'll go back for a longer time,'" Duntze said. "They were supposed to stay there for five weeks this time, it was the trip of their dreams." David Steiner was a business manager in a communications agency.