A Police Scotland spokesperson said: “Thanks to everyone who shared our appeal.”
No details about how Dobbie escaped or where he was found have been issued.
NHS Tayside previously said: “Due to patient confidentiality, we cannot discuss matters relating to an individual patient.”
Dobbie – who has also gone by the name Clifford Lyons and by Dobbie – was admitted to Carstairs after the attack on the 10-year-old girl.
At the time he was on bail for another sex offence – molesting a 12-year-old boy after luring him into a field and forcing him to strip.
Secure facilities, such as Carstairs, are used to house people who have committed murder, rape and other serious offences but have been judged by doctors to be mentally ill and in need of treatment.
In 2011 Dobbie brought a judicial review at the Court of Session in Edinburgh against a plan to ban junk food to combat inmate obesity at Carstairs.
He won the legal bid after he successfully claimed he needed high energy drinks and protein bars for his fitness regime.