The land was owned by the Catholic Church but was sold to the Ulster Wildlife Trust and incorporated into Bog Meadow Nature Reserve.
Campaigners said the land was used to bury babies born without being baptised and adults who killed themselves.
It is understood the deal was finalised after compensation was agreed.
Brilliant
Last year, the Ulster Wildlife Trust said it had agreed to the move in principle, subject to discussions on compensation.
The transfer is now expected to be legally completed in the next couple of weeks.
Last year, the Bishop of Down and Connor Noel Treanor visited the cemetery to consecrate the area it is believed was used as a burial ground.
It is thought that there might be as many as 11,000 people interred.
Donna Hanvey, who believes her baby brother was buried in the land after his death in 1966, said the latest development was being welcomed by the campaigners.
"We are over the moon. We knew that it was a matter of agreeing a figure of money and we thought they were at stalemate. But it's virtually done, so it's brilliant," said Ms Hanvey, who began her campaign for the consecration and transfer of the land in October 2008.
Remembrance
She added that work to transform the land was already planned, with work on a remembrance garden expected to begin within a short time of the transfer taking place.
In a joint statement, the Ulster Wildlife Trust and the trustees of Milltown Cemetery said they wished to thank all who those who had been involved in the discussions.
They expressed particular gratitude for the relatives of children buried in the cemetery.
They said that anyone with any queries about deceased loved ones should contact cemetery officials.
Bishop Treanor will attend Cemetery Sunday at Milltown today.
After prayer at the top of the graveyard, he is expected to make his way towards the Bog Meadow area at the bottom where he will hold a separate service for those buried there.
SIC: BBC