Portiuncula University Hospital in Ballinasloe, Co Galway, delivered 1,300 babies last year.
It serves east Galway, Westmeath, north Tipperary, Roscommon and Offaly.
It was previously the subject of reviews into the care of babies born there between 2019 and 2023.
At that time, eight reviews took place into cases of concern, with seven of these involving babies who had hypoxic ischaemic encephalopathy (HIE).
This involves a lack of oxygen supply to the brain before, during or after birth.
Six of the babies needed neonatal cooling.
This cooling treatment can last several days, to slow the baby's brain metabolism, to prevent or limit damage. The sooner the body cooling starts, the better the outcomes for the baby.
After these events, the HSE met with each of the families and shared the reviews.
At the time, the HSE said a series of improvement plans and other measures to deal with issues that arose would be put in place.
Also in 2015, a report commissioned by the HSE, and published in 2018, criticised the hospital for poor communication among maternity staff at the hospital.
The report raised similar concerns over the birth of eight babies where seven were sent for head cooling.
Now similar concerns have surfaced.
These cases involve two stillbirths in 2023, six babies born last year and one case this year.
The current status of seven of the babies who were delivered is not known at this time.
The HSE has said that last year, five of the babies were referred for head cooling, while there was another case that probably should have been.
The latest reviews have been set up after the incidents of HIE was found to be significantly higher than that observed nationally or internationally for a similar timeframe.
The HSE has apologised to the women and families for whom this news will be upsetting and worrying.
A helpline for women attending the unit who may have questions has been set up at 1800-807-008.
The HSE said once the reviews are completed, it will meet with each family to share the review finding with them.
While the reviews are under way, the HSE said all services will continue to operate as normal at Portiuncula.
A management team has been appointed to oversee maternity services at the hospital over the coming months.
It is being led by external consultant obstetrician, Dr Mark Skehan of University Hospital Limerick.
The team will be responsible for managing and supporting all aspects of maternity and gynaecology and neonatal services.
The new Minister for Health Jennifer Carroll Mac Neill has said her thoughts are in the first instance with the women and their babies who are the subject of the reviews.
She said that patient safety is paramount and once the reviews are complete, she will consider the next steps with her department officials and the HSE.
The HSE has said that the biggest concern is for women who have had their babies at Portiuncula and for those who will have their children there in the future.
These will be concerning times for women and their families who are due to have their babies at Portiuncula.
