A MUM who killed one of her twin baby girls and left the other severely disabled will not spend any time in jail.

The woman, 39, pleaded guilty to infanticide over the death of her eight-week-old daughter and recklessly causing serious injury to the baby’s sister.

Infanticide applies for mothers suffering a mental disturbance caused by childbirth, and carries a maximum jail term of five years.

She can’t be named, despite the father of the babies urging the court to lift a suppression order on her identity.

The father said he wanted the order lifted so he could talk openly about his lost daughter, who had “just disappeared”.

He also wants to raise much-needed funds for the surviving twin, he told the court in a statement.

But the Supreme Court heard that identifying the woman would not be in the best interests of her surviving children.

Justice Bernard Bongiorno refused to lift the ban on publishing the names or pictures of the woman and the twins.

The twins’ father, who was not present at two previous court hearings because he was at hospital appointments with his daughter, cried as the mother was sentenced to a community corrections order for one year.

Outside court, the twins’ paternal uncle — wearing a doubled-over pink lapel ribbon — said the sentence was “manifestly inadequate”.

The father had asked the judge to “please hand down justice for my daughters,” in a moving victim impact statement read to the court.

He said he was now acting as both mother and father to the couple’s older child and surviving daughter, who could not walk or talk as a result of permanent brain damage suffered at the hands of her mother.

The court heard the surviving children — the twin girl and a boy aged four — lived with their father, but their mother was still a legal guardian.

Justice Bongiorno said the father was awoken by noises made by his distressed, pale and limp baby girl, who stopped breathing, in the early hours of April 26, 2012.

Royal Children’s Hospital staff found a massive skull fracture had caused bleeding around her brain and the baby girl died on the operating table.

Medical staff found both girls had multiple historic and recent fractures to their skulls, ribs and other bones and bruising over their bodies.

Their mother had told a maternal health nurse during multiple visits she could not explain the twins’ visible bruises, claiming they may have been self-inflicted, caused by the other, or by their brother.

“They are like Houdinis — they get out of their wraps no matter how tight I wrap them,” she once told a nurse.

The babies, who had frequent crying bouts, trouble feeding and were not gaining weight, were diagnosed with colic.

After exonerating the baby’s father early on, police discovered his wife had been abusing the babies from the time they arrived home in late February following a caesarean birth.

Defence lawyer Peter Morrissey SC said his client told police post-natal depression may have caused her to “flip”, as she did not remember when or how often she hurt them.

He said his client told police: “They were just crying all the time and she didn’t feel worthy of being a mother because she couldn’t help them.”

The court heard she could not detail how she injured the babies, but once stopped herself when she realised she was shaking one girl, illustrating how she would rock them to the police.

She told police: “Maybe I shook them too hard like that.”

Justice Bongiorno said the woman’s mental disorder, connected with the recent childbirth, meant she could not be guilty of the charges of murder or attempted murder.

“It was clear they were inflicted by a loving mother suffering from significant emotional and psychological compromise,” the judge said.

The Supreme Court heard during a plea hearing last week that the woman was prone to depression and had an appointment to see a psychiatrist five days after her daughter died from injuries inflicted by her.

Justice Bongiorno agreed that there was nothing the court could impose that would be worse than the mother’s current suffering.

Mr Morrissey said the mother was a “broken” woman who suffered from an “acute level of self-hate, loathing, sadness and grief” and visited her daughter’s grave daily.

He submitted she should be pitied, not jailed.

“As for punishment, our submission is there is nothing more to be done — she has an ongoing sentence in that respect,” he said.

“She very much repents what she did and will live with that very much for the rest of her life.”

Mr Morrissey said no Victorian women had been jailed for infanticide.

The woman, who has been on bail since being charged in August 2012, remained on antidepressant medication and was a risk of suicide, the court heard.

Mr Morrissey said she was motivated by the faint possibility of being able to have contact with her surviving children in the future.