It is understood that the baby was overdue when the mother went into labour and it was decided to carry out a Caesarean section.
However, another C-section was already being carried out in the main theatre and a second theatre had to be readied.
When the baby was born, he was critically ill and was rushed to the National Maternity Hospital in Dublin but died shortly afterwards.
The death of the baby comes as the HSE and the Government face increasing pressure from the fall-out over the death of Savita Halappanavar in Galway after a miscarriage.
Today Savita's husband Praveen has criticised the Galway Hospital group for taking almost a month to express its sympathies to him and Fine Gael TDs are voicing strong opposition to the introduction of laws that recognise suicide as grounds for abortion.
In relation to the Cavan baby death, a HSE spokesman said: "The HSE wishes to express its deepest sympathy to the parents of the deceased infant. The HSE has offered support to the parents at this time.
He added that "as is normal procedure the HSE will closely examine the circumstances of this case".
"A clinical incident review is being planned which will involve a multi-disciplinary team reviewing all aspects of the care provided in this case."
About 160 babies are born at the busy general hospital every month. The number of births had double in the last 10 years with a total of 1,431 births at the hospital for the first mine months of this year.
Ireland currently has one of the lowest infant mortality rates in the world with 3.81 deaths per 1,000 live births which is lower than the UK at 4.56 and the US at 5.98 per 1,000 births.