The Government has agreed to draft new laws, allowing for the excavation of the former Tuam Mother and Baby Home site in Co Galway.
The legislation will also allow for the remains to be exhumed and DNA testing to be carried out to try and identify the babies.
It has been confirmed the Sisters of Bon Secours, who ran the home, will pay €2.5m towards the cost.
However the full cost could be up to €13m.
The Children's Minister Katherine Zappone says the bill will create the legal basis for the course of action that Government decided should take place at the site.
Under the bill, there would be a legal basis to carry out a programme of phased, forensic standard excavation, exhumation and re-interment of remains at the site.
There would be a legal basis for a programme of forensic analysis of any remains, with samples being taken from the remains as well as from family members of the deceased.
Bones and a shoe were among items discovered in the tanks of the former home back in April.
An agency is also to be established to "manage intervention at the site and act as a dedicated and responsive authority."
The Department of Children and Youth Affairs says any such agency would operate for a specific and limited duration - and would cease to exist as soon as the work was complete.
The Government could also authorise similar interventions at current or former institutional sites other than the Tuam site, if it is deemed necessary in specific circumstances.
"This is notwithstanding the fact that, from a legal and ethical perspective, it is considered appropriate to leave burials undisturbed wherever possible", the department adds.
"In this context, therefore, a burial site must be manifestly unsuitable and meet other certain defined criteria in order for Government to consider an intervention."
Minister Zappone says: "The drafting of this legislation is a priority for me, and I am delighted that Government supports the approach outlined in the general scheme to deal with the various sensitive and complex issues at hand.
"I know that family members of the children interred at the site in Tuam have been eagerly awaiting this development and I am pleased that agreement on a general scheme marks a significant milestone on our journey to afford those buried there the dignity and respect that they deserve.
"I am also mindful that local residents will be impacted by action taken at the site and I am confident that the general scheme strikes an appropriate balance between doing what is right for the deceased, while minimising negative effects for those living nearby."
The text of the general scheme can be found here