Nicola Sturgeon said the authorities cannot be sure there are no other Scots affected.
Compensation In Ashes Scandal
A compensation scheme to settle claims from more than 100 families affected by the Mortonhall baby ashes scandal has been offered by the council, which also announced two memorial sites are being planned.
Over decades, the remains of stillborn and dead newborn babies were disposed of at the Edinburgh crematorium without the families knowing.
Parents were instead told there would be no ashes left to scatter following cremation.
A year-long inquiry into the crematorium's practices from 1967 to 2011 concluded that the situation was a "great tragedy'' which left many parents facing uncertainty about their baby's final resting place.
Edinburgh City Council announced today that a proposed settlement scheme developed with lawyers representing 129 families will provide payments of between £1,000 and £4,000 - avoiding the cases being pursued through the courts.
Council officials also revealed designs for two permanent memorials to be developed later this year, with one planned for Mortonhall.
It was decided a second memorial should be built elsewhere in the city as some families said they would never return to the crematorium grounds. Suggested sites include the city's Princes Street Gardens and the Meadows.
A survey involving affected parents was carried out over what the memorials should look like, with responses including a sculpture, a statue or water feature.
The draft designs are contained in a report going before councillors next week when they will decide on the proposed compensation scheme.
It also outlines the progress made by a multi-agency working group set up by the council to take forward recommendations in Dame Elish Angiolini's 600-page report published last April.
Dame Elish was appointed by the council to head the investigation into former practices at Mortonhall after they were uncovered in 2012 by child bereavement charity Sands Lothians.
Her final report centred around the cases of more than 250 babies.
A separate independent commission, led by former High Court judge Lord Bonomy, was later set up to review policies and practice across Scotland in relation to the handling of ashes following the cremation of babies and infants.
His report, published in June, produced 64 recommendations aimed at improving policy and practice across Scotland, with a new national investigation team created.
Sue Bruce, chief executive of Edinburgh City Council, said she was "very encouraged'' with the progress made by the group in dealing with the recommendations from both reports.
She said: "I will take the paper to council next Thursday and recommend we proceed with this settlement.
"The questions that were raised about practices at Mortonhall at the end of 2012 relating to practices over a long number of years have raised some really difficult questions. It's been very, very difficult for the affected parents and has reopened grief.
"I think for us as a public authority to try and reach a reasonable settlement without a prolonged, difficult process is important.
"It is showing a sign of respect to the parents concerned and there is no need for us to make it any more difficult than it has already been.
"It has been a shocking turn of events and I take this opportunity to reiterate once again my apology on behalf of the council that parents have had to go through this, and I hope that they see with the commitment to implement the recommendations, to work with parents, to work with government and to work in future with the independent inspector of crematoria, shows a commitment by the council that, certainly in Edinburgh, this will not happen again.''
Dorothy Maitland was operations manager of Sands Lothians when the scandal was uncovered.
The 59-year-old, who has since retired from the role, had her daughter Kaelen cremated at Mortonhall in 1986 and was told there would be no ashes to scatter.
She said: "I thought I would never be able to go back to Mortonhall. I felt so angry and so hurt, and I just thought it would never be a place I would want to go and remember my daughter.
"But I think as time went on, I thought that was where we left her and my feeling now is that it's important there is somewhere I can go if I want to just to go and reflect.
"But I think it's very important that there is another location because there really are parents who can't go near Mortonhall - they can't even drive past the place. Their needs have to be met.
"I see the settlement as the council saying they did us wrong and they want to try and apologise. This is their way of saying that they do care and want to show that they do.
``I think, and I am hoping, that a lot of parents will be happy they've put their hands up and admitted it, and that parents can find a bit of closure.''
She added: "I will never be able to have my daughter's ashes but because of her and other babies there are going to be huge changes and this will never happen again, and that's a huge thing. That for me gives me the most comfort.
"Standing here today I am very happy of the outcome and I can't praise the council enough for how much they really listened, and that was so important.
"I am very proud of every parent that let our voices be heard and more importantly every baby - it's because of them that we are standing here today and I think that what drove us all the most was the memory of our babies. That makes me feel proud and happy.''
Willie Reid, 48, whose baby daughter was cremated at Mortonhall, said he initially thought he would be unable to go to back there but welcomes a memorial being built.
He added: "I've reflected on it and that was the last place I had Donna in my hands, I carried her down the chapel, so I think the fact there's going to be a memorial for her and all the other babies, I think a change of mind has been right.
"But there has to be a second memorial because there are some parents who will never return to Mortonhall and they have to be catered for as well.
Talking about the settlement, he said: "I think it is something that has to be welcomed because it brings closure to any legal situation.
"I'm not in this for any money, the very fact that the legal side of life is changing, that's the legacy of my daughter - so the money is not an important thing for me.
"I think of my daughter now and I think 'you've changed something for the short time that you were on this earth, you are now going to make a massive impact'.''
Patrick McGuire, from Thompsons Solicitors, who represents the 129 families, said: "As a group, the Mortonhall parents have displayed great solidarity and compassion to each other, and it is largely through their efforts that we now have wider investigations into events at crematoria throughout Scotland.
"The proposed settlement scheme makes very clear that none of the families we represent will be excluded and again this is testament to the strong will and dignity shown by my clients.
"The Mortonhall families, through their tenacity and decency, have brought us to where we are today.
"In doing so they have ensured that in the future no other families will have to suffer distress due t unacceptable practices at our crematoria.''
Family members say Laura Macintyre and Eilidh MacLeod haven't been heard from since the explosion.
The families of two girls from the Western Isles missing after attending the concert in Manchester targeted in a terror attack have issued appeals on social media.
Crews were called to the former Royal Victoria Hospital site this afternoon.
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