The 32-year-old marketing executive became pregnant after a brief affair with a man she met at work.
She kept the pregnancy a secret from her partner David Stoneman and told him she was putting on weight because of a wheat allergy.
On the night of December 27 last year Jones gave birth alone in the bathroom of her mother's home, but it proved stillborn. She might have thought it was a blessing in disguise, but it wasn't, as it turned out later.
She wrapped the stillborn baby in a carrier bag and some plastic bin bags then drove back to the house she shared with Stoneman, 33, Daily Mail reported.
Jones acted as if nothing had happened and even returned to her job as a marketing manager.
But South Wales Police were tipped off by a health service worker and called at Jones's home in St Mellons, Cardiff, 10 days later.
She told officers the baby was stillborn and she had flushed it down the toilet.
But forensic officers searched the house and her car and found the dead baby concealed in the boot.
Her partner Stoneman was questioned initially but was later released without charge.
Two post-mortem examinations of the child's body were carried out, including one by a Home Office pathologist.
Jones, who said throughout that the baby was stillborn, was later charged with endeavouring to conceal the birth of a child in December last year.
In the UK it is a crime not to inform the appropriate authorities of stillbirth or death of newborn. Under the Offences against the Person Act of 1861, it is punishable with up to two years of imprisonment.
Jones will be sentenced next month and was remanded on conditional bail to her mother's home and the judge ordered that the address is kept secret.
Judge John Curran said: 'Nothing should be read into the fact bail has been granted today.'
Stoneman has left the couple's home in St Mellons and the couple are no longer together.
A neighbour said: 'It's all very sad. They were a nice couple and Claire had a good job.
'When she put on weight last year she told everyone she had been diagnosed with an intolerance to wheat.
'No one questioned it and she said she was getting treatment which would sort it out.'
It is understood Jones was seen by a doctor when she suspected she was pregnant in May of last year.
Her fears were confirmed but she failed to attend ante-natal clinics or appointments for scans and other tests.
A South Wales Police source said: 'We were called in because there was a record of a woman being pregnant but no record of her baby being born.
It is a very sad set of circumstances.'