Keith Pollock assaulted the 16-year-old stranger as she made her way home in the early hours of January 26 this year.
He grabbed the teenager and punched her before trying to force her into a darkened park. She had been on the phone to her boyfriend, who heard her screams before the line went dead.
Pollock eventually fled when a delivery driver heard the victim screaming for help and confronted him.
The 50-year-old was later traced and the girl picked out his unusually deep voice in a voice-identification parade.
Pollock is now back behind bars after he admitted assaulting the teenager with intent to rape, at the High Court in Glasgow yesterday.
It emerged Pollock had been jailed for seven years in 1991 in Australia for rape, public indecency and burglary.
He was deported after being freed from the sentence and was living with his partner and young child in Milton of Campsie, East Dunbartonshire, when he attacked the teenage victim, who cannot be named for legal reasons.
Pollock now faces another lengthy spell behind bars when he is sentenced in July.
The court heard how the girl had left her boyfriend's house in Milton of Campsie in the early hours and decided to walk home.
She later spotted Pollock lurking nearby before he ran up behind her.
The teenager called her boyfriend and told him she believed she was being followed.
Prosecutor Ian Wallace told the court: "Her boyfriend heard nothing for a few seconds and then he heard his girlfriend screaming. At that point the phone cut out."
Mr Wallace explained how Pollock grabbed the girl, put his hand over her mouth and tried to drag her backwards.
He then repeatedly punched his victim in the face before attempting to push her over a wall into the park.
The advocate depute told how the victim "screamed throughout the attack" and believed she was going to be raped.
Pollock slapped the girl in a bid to silence her - but in the meantime the delivery driver had heard her crying for help.
The man confronted Pollock, who told him "I'll f****** kill you" and then ran off.
The injured girl was found and police began the hunt for Pollock.
The court heard how on March 1 the victim took part in a voice-identification parade.
She had recalled her attacker having "a really deep voice", which led to her picking out Pollock.
His DNA was also found on the girl's top, leading to his arrest.
The court was told the teenager is now paranoid about going out alone and afraid of people walking behind her.
Pollock was remanded in custody as Judge Lord Burns deferred sentencing for reports.