Kamiyah Mobley, who was abducted in July 1998, was found after a tip.
Authorities in Walterboro, South Carolina, have charged Gloria Williams, 51, with kidnapping.
Ms Mobley was living under another name and believed Ms Williams to be her mother. The biological family have been notified of the news.
Jacksonville Sherriff's Office said DNA tests had confirmed Ms Mobley's identity.
She appeared in good health, a "normal 18-year-old woman", it said.
The office said it had acted on 2,500 tips since the abduction and received one last year to @MissingKids that eventually broke the case.
It showed the original composite images released at the time of the kidnapping, along with a photograph of Ms Williams, who was arrested at her home on Friday morning.
The baby was only eight hours old when she was taken by a woman posing as a health care worker at the University Medical Center, now known as UF Health Jacksonville.
The woman told Kamiyah's mother, Shanara, that her baby had a fever and needed to be checked. She took the baby out of the room and disappeared.
The case brought significant media attention as Shanara tried to locate her child.
Although the biological family had been notified of the discovery and were "elated", the Jacksonville Sherriff's Office said: "It is up to the victim on how contact will be made. The victim is now an adult."
Jacksonville Sheriff Mike Williams said: "She had an idea that something may have occurred but... imagine her trying to process this.
"She was abducted as a newborn and needs time to process this...  We want to respect her privacy and we ask that you do too."
A girl stolen as a newborn from a hospital in Jacksonville, Florida, has been found alive in South Carolina after more than 18 years, police say.