The HSE said the archived cards, which are more than 10 years old, will be disposed of to meet Data Protection Regulations. However parents can apply for their return.
An HSE spokeswoman said: "Over one million cards are in storage for 1984 to 2002."
However, the Royal College of Physicians has expressed concern at the decision, saying: "This will result in the permanent loss of a genetic database of the Irish population.
"We do not agree that the State is obliged by European law to destroy this natural archive. Destruction of this priceless national asset is the easy but wrong option."
Babies are screened for six rare conditions soon after birth, in what is known as the heel prick test. It involves taking a few drops of blood from the baby's heel for testing.
This sample is tested for a number of conditions including: phenylketonuria, maple syrup urine disease, homocystinuria, classic galactosaemia and congenital hypothyroidism.
Screening cards from babies born between 1984 and 2002 will be disposed of this year, the HSE said.
"A tender process will be conducted to ensure this is done securely, confidently and appropriately."
Dr Kevin Kelleherof the HSE said: "During 2011, the information and consent processes for parents was enhanced.
"Parents whose child is having the test now give consent for the test and for the card to be stored for up to 10 years.
"The HSE has been asked to ensure that newborn screening cards that are more than 10 years old, and date from the 1980s to 2002, are disposed of.
"This is because no explicit consent for storage was given at the time these tests were done, and therefore data protection legislation requires the HSE to not continue to store them."
Any person who prefers that their card is not disposed of can have it returned to them by March 31.
The HSE said: "Some Irish health research and patient organisations have expressed an interest in accessing and storing newborn screening cards to further their research into health conditions."
It said it has invited a range of organisations to submit details of governance arrangements for such storage, or research uses for cards.