The Bishop of Down and Connor, Noel Treanor, said the Trustees of Milltown Cemetery regretted "the hurt and distress" the move caused.
The plot was leased to the Ulster Wildlife Trust in 2000.
Donna Hanvey, whose baby brother Michael is buried on the land, said she was "over the moon" at the news.
"It seems to have lifted part of the anger that we've had about what happened to these babies and I don't just mean our relatives but all the babies," she said.
On Friday the Trust said it would help return the land "as quickly as possible".
Bishop Treanor said there were plans to erect "an appropriate monument" to the memory of those buried at the site.
The UWT's Chief Executive Heather Thompson said that they would fully co-operate in the return of the land.
"The Ulster Wildlife Trust is working closely with the Church's representatives to ensure the land transfer process is completed as quickly as possible," she said.
In an apology, printed in Belfast newspaper the Irish News, the Bishop said the land had been leased "in good faith".
It is thought that hundreds of graves may be contained within the plot of land, some dating back to the 1860s.
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(Source: BBC)