The girl made emergency calls to Wiltshire Police from telephone kiosks on 16 and 23 November, reporting that a non-existent fight was in progress.
She was identified when officers reviewed CCTV of the kiosks in Calne and posted images of her on Facebook.
She was invited to Calne police station where she admitted both offences.
Police said the first call was made from a kiosk in Church Street. When a PCSO attended the scene he found nobody there.
The second call was made from a kiosk in Phelps Parade, but again nobody was there when officers attended.
There were many responses from people to the CCTV images of Church Street posted on Facebook and, during the appeal, the girl also contacted officers to tell them what she had done.
A further review of the footage from Phelps Parade established the same girl was responsible.
She was invited to the station with her mother and questioned about both offences.
As the girl had not been in trouble with the police before, and because of her age, it was agreed to resolve the matter with a so-called Local Resolution.
She was told to write a letter of apology and then attend force headquarters to give her apology in person.
A 13-year-old girl who made hoax calls to police has been made to read an apology to the 999 call handler supervisor.