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.
The Irish health authorities have begun an investigation following the death in hospital of a newborn baby.