Maggie Oldham from the Mid Staffordshire NHS Foundation Trust, read a statement today.
She said: "Our deepest sympathies remain with Alfie's parents and their loved ones and we are very sorry that the care provided was simply not good enough."
Richard Follis, Family solicitor said that there was no doubt that the morphine played a part in the twins' deaths.
He said if the morphine had not been given, probably both twins would be alive today.
Mother Ami Dean was taken to Stafford Hospital on 30th October 2010.
The twins she gave birth to were 13 weeks premature.
After their condition deteriorated, they were transferred to University Hospital of North Staffordshire but died on November 1st.
Twins Alfie and Harry McQuillan died after being given more than ten times the recommended dose of morphine at Stafford Hospital.
Read the full storyParents of Alfie and Harry McQuillan describe what happened when they discovered the overdose of morphine had been given.
One twin was given ten times the recommended dose while the other twin was given 17 times the normal dose.
Both died soon after the drug was given.
An inquest into the deaths of twin baby boys who died in Stafford Hospital has recorded a narrative verdict.
Alfie and Harry McQuillin were born 13 weeks early at Staffordshire Hospital.
The coroners' court was told how they were given more than ten times the recommended dose of morphine, by a nurse who had little understanding of how to do it.
The coroner recorded a narrative verdict and said that they died of complications of prematurity.
The coroner said the excessive dose of morphine was a likely factor for their deaths and admitted there had been failings in their care.
Stafford Hospital has apologised to the parents.
An inquest into the deaths of twin baby boys who died in Stafford Hospital has recorded a narrative verdict.
Premature babies Alfie and Harry McQuillin died after being given ten times the amount of morphine they needed last October.
More to follow.
