The baby died about six weeks ago but few details of the case were released.
The child was born in Midland Regional Hospital, Mullingar, County Westmeath, on 16 August and died on 22 August at Dublin's National Maternity Hospital.
The Health Service Executive (HSE), which runs the Republic of Ireland's public health services, confirmed it was investigating the "tragic" case.
In a statement to the BBC, a spokesman said: "The HSE incident management policy is being followed and a full and thorough investigation into this tragic event has commenced.
"The family are involved in this process. The hospital extends its deepest sympathy to the family," he added.

Baby Max Naman died at the scene of the crash in Finistere on 7 June. His father, Lawrence Naman, 37, of Mile End, died the next day in hospital.
Opening inquests into the deaths coroner Caroline Beasley-Murray said Mr Naman's wife remained in hospital.
The hearing, in Chelmsford, was adjourned to a date to be confirmed.
The court heard how Mr Naman had been driving the car with his wife - who was not named in the hearing - by his side and baby Max was in the back when their vehicle collided with a lorry.
Mr Naman was taken to Brest Hospital where he later died. A post mortem examination found he died of skull trauma.
Mrs Beasley-Murray said the deaths were a "tragic loss" and were currently subject to a criminal investigation in France.
It is understood the inquest will resume once the police investigation has been completed.
The Irish health authorities have begun an investigation following the death in hospital of a newborn baby.
[NEXT_CONCEPT]
A father and his nine-month-old son from east London died in a road crash during a family holiday in Brittany, France, an inquest has heard.