Abbey Cape is in a "stable but critical" condition
Abbey Cape, from Ashby de la Zouch in Leicestershire, had been on a life-support machine since 13 April.
Her mother, Marianne Cape, 28, and stepfather, Darren Jackson, 32, made a nationwide plea for a donor to come forward.
A donor was finally found and Abbey had the transplant "within the last few days", according to the hospital.
'Critical time'
A spokesperson confirmed on Monday that she was in a "stable but critical" condition in the transplant suite in the intensive care unit of the London hospital.
In a statement, Ms Cape and Mr Jackson said: "Doctors warned us that the first 48 hours post surgery would be a really critical time for Abbey.
"We are delighted that a heart has been found for her but realise that it is still very early days."
They said they were grateful for the continuing support they had received but asked for privacy during a "difficult time" for them.
Abbey's parents first became worried about her five weeks ago when the youngster suddenly became weak and unable to hold down food.
After tests at Queen's Hospital in Burton upon Trent, she was sent to Birmingham Children's Hospital.
Support messages
There she was diagnosed as suffering from severe cardiomyopathy - an infection of the heart muscles which causes them to swell and prevent blood from flowing normally.
She was then transferred to Great Ormond Street where doctors said her only chance of survival was a transplant operation.
Last week, children at Abbey's school, Snarestone Primary, sent tapes of support to her.
Fund-raising events have also been organised by people living in the town.