The sheriff did not give any details on how Elizabeth died or about the juvenile in custody except to say that the person lived in the area west of Jefferson City and was older than the girl. Police said Elizabeth's body was found just before 3 p.m.
"We were able to obtain some physical evidence and through some analysis of some of the evidence and in all honesty some written evidence, we were able to develop a person of interest," White said. "Once we reached that person and interviewed them, ultimately they led us to where we've recovered Elizabeth's body."
Elizabeth, of St. Martins, was last seen when she started walking home from a neighbor's house on Wednesday evening, police have said.
White said had spoken with Elizabeth's mother and said the family is "deeply grieving."
"I'm a parent, and I know how I would feel. I would simply leave it to you that they are grieving," he said.
Elizabeth's family and friends have described her as a bashful fourth-grader - the youngest of five children - who loved cats but was afraid of the dark and would not normally have gone into the woods. Some of her family members had spent Friday making buttons with Elizabeth's photo, before learning of her death.
"She was a wonderful, happy, shy child. I never saw her argue with her mother, her sister or anyone," said Liza Adrian, the sister-in-law of Elizabeth's sister, who spoke on behalf of the family during a Friday night news conference.
Elizabeth's father, Dale Olten Sr., learned of her death while watching TV in prison, where he is serving a four-year sentence for drug possession, said family friend Nancy Hanks, who had aided in the search.
"On behalf of the Oltens, I guarantee they appreciate this community - all the outpouring of support - they worked hard" searching for Elizabeth, Hanks said.
Elizabeth's aunt, Vicki Olten, described her niece as an "angel" but declined further comment.
Earlier Friday, about 70 people had searched for Elizabeth on horseback, with all-terrain vehicles and on foot while another 70 investigators checked leads, White said. Several hundred people had joined the search Thursday despite a steady rain and rough terrain.
Police focused their search area after they figured out roughly where Elizabeth's phone was located. White said the phone was later found but declined to say if it had turned up evidence.
The Missouri State Highway Patrol searched by helicopter with thermal imaging radar, and the Missouri State Water Patrol checked ponds in the area with sonar.
Highway Patrol Superintendent Col. James Keathley said he wishes the outcome was better.
"It's been quite an ordeal for the last few days. There's a lot emotions involved in this. It's been tough on everybody involved in this case," Keathley said.
David Schulte, who lives nearby, said Elizabeth typically would walk through his front yard about 30 or 40 feet away from the road on her way back home from the friend's house. Schulte said he didn't know Elizabeth well but that she had come through the neighborhood to sell cookie dough for a school fundraiser.
Schulte, who had helped with the search, said the area around where the phone was transmitting signals was about one-quarter of a mile away from the county highway and in the middle of the woods. He said it is easy to get turned around and difficult to hold straight search lines because the terrain is all "ridges and valleys" along with brambles.
"You'd have to want to be back there," said Schulte, before police announced that Elizabeth's body had been found.
Cole County Prosecutor Mark Richardson did not immediately return a call seeking comment Friday.
Associated Press writer David A. Lieb contributed to this report.