The six-and-a-half-hour operation on the two-month-old twins was carried out by one of the world's most experienced teams, including Lewis Spitz, professor of paediatric surgery.
He has previously operated on six sets of twins, of which seven children have survived.
The twins, born on 7 April, were joined at the chest and abdomen but doctors were hopeful they could separate them after it was discovered they only shared one organ - the liver.
The girls were transferred to Great Ormond Street at the beginning of last week and preparations for separation began, including a series of detailed X-ray investigations, to determine the extent of organ-sharing.
Although they had separate hearts, the pericardial sac, which surrounds the heart, was shared. Surgeons were able to reconstruct one pericardial sac but a prosthesis was needed for the second twin, as there was insufficient tissue to construct a separate sac around her heart.
With the liver there was a considerable area of fusion, the hospital said. It was divided using an ultrasonic dissector, tissue coagulation and medical glue to prevent bleeding. In fact, little blood was lost.
The procedure was carried out by Professor Spitz, Edward Kiely, consultant paediatric surgeon, and assisted by Alan Dickson, consultant paediatric surgeon from St Mary's Hospital, Manchester. Four anaesthetists and a team of doctors, nurses and operating department assistants were involved.
The operation began at 9am and finished at 3.30pm. The twins were then transferred to intensive care where their condition was described as "stable".
"These operations are always extremely complicated and despite a very detailed range of investigations beforehand it is impossible to determine exactly what to expect on the day of the operation," said Professor Spitz. "Today's operation, however, went very well.
"It is too early to say how the twins will fare; they are very young and have undergone major surgery. They are making good progress so far. We are keeping a close eye on them in the intensive care unit and are doing all we can in the hope they will both pull through."
Great Ormond Street Children's Hospital is at the forefront of treating Siamese twins. Two years ago the team led by Professor Spitz and Mr Kiely operated on Siamese twins Chloe and Nicole Astbury. They were joined from the breast bone to the navel and underwent an operation to separate their bowels in the first few weeks of life.
Complete separation was planned for sometime in their first year but the girls died from a bowel disease, which spread rapidly to other organs at the age of six weeks.
The best known recent case of Siamese twins is that of Eilish and Katie Holton, who were born in Ireland in 1988, and whose lives to the age of three were the subject of an acclaimed documentary.Reuse content