Debra , who is now 40, explained they began IVF treatment in 2000 following an ectopic pregnancy which had damaged a fallopian tube.
On their third attempt, the couple's son, now six, was conceived. Debra said they would call him "our little miracle".
The remaining embryos were frozen and, in line with the clinic's policy, were kept for five years.
In November 2007 the clinic contacted the couple with the news that just one embryo had survived and was in good condition.
The couple said that they arranged an appointment at the IVF Wales clinic for the following month. Debra was gowned up waiting to undergo the operation when the couple were told there had been an accident in the lab, and they could not go ahead.
Her husband Paul spoke of their "shock and anger" and said and went home from the hospital immediately.
This is an unacceptable accident and should not have happened and I would like to apologise to the women involved and their families for our failings in this case.
Ian Lane, Cardiff and Vale NHS Trust
"In less than 10 seconds our wonderful world was shattered when the senior embryologist stood in front of us and said, 'I'm very sorry to tell you, but there's been an accident in the lab. Your embryo has been destroyed'," she said.
"We were both rooted to our seats. We were stunned and trembling. We held each other tightly, and sobbed and sobbed."
In a later interview, Paul said: "At the time there was no real explanation. It was just that the embryo had been accidentally implanted into another couple."
He said they had fought through their solicitors for a full investigation because their major concern was that no other couple should have to go through what they had done.
He said they had been offered free treatment by the hospital but declined after what had happened.
Ian Lane, medical director of Cardiff and Vale NHS Trust said: "This is an unacceptable accident and should not have happened and I would like to apologise to the women involved and their families for our failings in this case."
Mr Lane said the incident was investigated internally and by the Human Fertilisation and Embryology Authority (HFEA.) and the issues identified have subsequently been addressed.
"The fundamental failing was a failure to witness on two occasions the identity of the embryo," said Mr Lane.
"There are other issues which contributed to this, one of which was the peaks and troughs of the work. And at the time this happened a working incubator was being used inappropriately.
"There was poor lighting, it was at lunchtime, and there was vibration occurring from adjacent building work."
Mr Lane said the trust has now reduced the number of embryo transfers taking place each week to dovetail them with staffing levels.
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