Traherne Williams hurled tiny Sophia Williams onto a couch causing catastrophic brain damage.
Sophia had been taken to hospital ill days earlier before being allowed home.
The 22 year-old later lied to a doctor the baby had fallen at a family centre they had been staying at in Broxburn, West Lothian.
Sophia never recovered and died in hospital in June 2016.
Williams and the child’s 20 year-old mum Shannon Main – the killer’s lover – were both later charged with murder. It led to the parents appearing at the High Court in Glasgow.
Williams pled guilty to the reduced charge of culpable homicide. Main walked free after prosecutors accepted her not guilty plea.
Williams was remanded in custody pending sentencing next month.
A judge was meantime told Main is standing by the man who killed their child. The couple started dating in 2014 before Main fell pregnant with Sophia the following year.
They initially stayed with William’s mum, but later ended up at the Strathbrock Family Unit in Broxburn, West Lothian.
This was just three days before Sophia was born.
The centre is a homeless unit – the couple were staying there while waiting for their own tenancy.
Sophia was then born on February 20 2016.
On May 28 – two days before the fatal attack – Sophia was taken to hospital as she had been sick for a number of days.
It was suspected she had a virus, but was deemed well enough to go home.
On May 30, the pair were at the unit with Sophia when Main tried to feed her.
But, she was found to be “less responsive than usual”.
She was later in her Moses basket and her parents could not rouse her.
Main contacted St John’s Hospital in Livingston after stating she had been “unresponsive for around an hour”.
Prosecutor Angela Gray said: “When paramedics arrived, it was immediately apparent the child was extremely sick.”
Sophia was rushed to hospital – it was initially feared she had meningitis.
Miss Gray: “On asking Williams whether the child had been injured, the doctor was told she had fallen from a couch three or four weeks previously.”
Williams was later described as “upset” - but medics felt he was being “defensive”.
A consultant believed Sophia brain injury was “highly suspicious of non accidental injury”
On June 4, Williams claimed he had to leave hospital due to “anxiety issues”.
Sophia never recovered and she was pronounced dead the following day.
She was found to have suffered a “traumatic head injury”.
A police probe soon began into Sophia’s tragic death.
The court heard Williams, now of Linlithgow, West Lothian, went with his mother to consult his lawyer on August 3 2016.
He made various remarks to her including: “Maybe I held her wrong”.
He was later quizzed by police and denied assaulting his daughter deliberately.
Miss Gray: “He stated he may have done something accidentally. He did not specify precisely what.”
The prosecutor said the guilty plea was accepted on basis Williams “became frustrated at his inability” to stop Sophia crying while alone with her on May 30.
He was then said to have “forcibly threw her onto a couch” in the living room.
Miss Gray: “The type of force used would be immediately apparent to anyone watching as being significant.
“The result of throwing would necessarily result in movement of the head and neck.
“This is due to the fact the infant does not yet have the capacity to be aware what is happening and to control the head.”
Ian Duguid, defending, told the court Williams found it difficult to understand how a “single reaction” could have “such horrendous consequences”.
The QC added: “He had effectively projected the child onto a sofa or soft surface not realising he had inflicted such a catastrophic injury in the process.
“He was a first time father obviously with little experience in looking after a child.”
Mr Duguid also said Williams relationship with Main “continues to this day”.
Lord Matthews deferred sentencing for reports until June 8 in Livingston.