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."

The Athens General Index opened down about 4%, before recovering to be 1.4% lower. On Monday, the share index had dropped 16% on the first day of trading after a five-week shutdown.
Major banks were close to the maximum 30% declines they are allowed in one day, a limit that was hit on Monday.
Banks account for about 20% of the Athens index.
Falling bank shares were offset by gains in some non-financial shares. Among the shares rising were the gaming group OPAP and some tourism-related shares such as Aegean Airlines.
The main share index is down more than 50% from last year's peak.
During the height of the Greek debt crisis, the stock exchange was closed for five weeks as part of the programme of capital controls to prevent people taking euros out of the country.
Greek investors are still not allowed to use money in their bank accounts to trade on the stock exchange.
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.
[NEXT_CONCEPT]
Greek bank shares have fallen sharply for a second day in Athens, but non-financial firms have fared better.