Mary O'Hare, 88, the dead girl's mother, and her son, Michael, met with Owen Paterson, the secretary of state for Northern Ireland, at Hillsborough Castle in Belfast, the Belfast Telegraph reported. Paterson gave them a letter from Defense Secretary Liam Fox.
Majella O'Hare was shot in the back as she walked to church. Michael Williams, who shot her, said he fired because of an IRA gunman in the area, and was acquitted of manslaughter in 1977.
The apology was in response to a report by a Historical Enquiries Team.
"The soldier's actions resulted in the loss of a young and innocent life, causing sorrow and anguish for those who knew and loved Majella," Fox said. "On behalf of the Army and the Government, I am profoundly sorry that this tragic incident should have happened."
Mary O'Hare told the Telegraph she tried to talk to Williams at his trial: "I went up to him in the court and asked him: 'Are you proud of what you did, killing Majella?' He just shrugged his shoulders."