NHS Lanarkshire has said "sorry" for the distress caused after a foetus was inappropriately discarded at Monklands hospital in Airdrie, North Lanarkshire, last week.
It added that an investigation has been launched.
Michael Coyle, councillor for Airdrie South in North Lanarkshire, described the situation as "appalling".
He said: "I think it's appalling in this day and age that something like this can happen. There's been no consideration for the feelings of the parents.
"I think it's absolutely shocking and whoever is responsible has to be sacked because this is unacceptable.
"Parents go through enough whenever they lose a child. It's a traumatic experience all-round and for a wee baby to get dumped into a bin, it's not acceptable.
"It's not a piece of rubbish, it's a life that's been lost."
Health Secretary Alex Neil, who is also the local MSP for Airdrie, has demanded that procedures be "tightened up" and steps be taken to ensure a similar incident does not happen again.
Mr Neil said: "Something has gone badly wrong with procedures, as a result of which a foetus was wrongly put in a repository that it shouldn't have been put into."
He also said the situation had clearly gone "seriously wrong" as the proper procedure in these circumstances is for foetuses to be stored at nearby Wishaw general hospital, also in North Lanarkshire, where they are kept in a "protected and respectful environment".
Mr Neil added: "The health board is doing two things.
"Number one, looking at whether the procedure needs tightened up, which clearly it does, to make sure this doesn't happen again, and secondly, obviously looking at why this happened in the first place, and presumably looking at whether disciplinary action of any kind is appropriate.
"Although the family hasn't contacted me as their MSP, very clearly it's a very, very distressing situation.
"I don't know the circumstances under which the foetus didn't go to full period, but clearly under any circumstances this is a real tragedy, particularly for the mother."
Mr Neil also referred to "other issues" at Monklands which have caused the hospital to come under fire previously.
In August last year it was discovered that patients had been wrongly marked as do-not-resuscitate, in 2011 seven wards closed as a result of an outbreak of the winter vomiting bug norovirus and in 2009 the mother-in-law of Labour MSP Michael McMahon was left on a trolley for eight hours waiting for a bed.
Mr Neil added: "This is the first time, to the best of my knowledge, that this has happened (at Monklands).
"We've had other issues around Monklands, but to be fair to NHS Lanarkshire they've been doing their best to make sure that Monklands' performance is brought up in terms of waiting times, in terms of A&E turn around, in terms of reducing c.diff and other hospital-acquired infections.
"There has been a huge improvement in Monklands over the past two or three years."
A spokesman for NHS Lanarkshire said: "We are sorry for the distress caused by this incident.
"It is our aim to treat all families with dignity and respect, and we are carrying out a critical incident review as a matter of urgency to establish what happened in this case.
"Following the outcome of this review we will ensure that all recommended actions are put in place across NHS Lanarkshire.
"We do have arrangements for sensitive disposal of a foetus. These are discussed with the family and vary depending on individual circumstances. Options include a hospital cremation or a private burial or cremation. These procedures have now been followed."
The critical incident review must be completed within 20 working days.
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