Both girls made it through the separation operation but Hope's lungs were too small to support her breathing.
Her parents Laura and Aled Williams were at her bedside when she died Tuesday. Doctors said the couple are devastated but are now praying Faith will pull through.
Laura, 18, had made medical history by going through with the birth to become Britain's youngest mother of Siamese twins.
Faith is breathing with the help of a ventilator and her condition is stable, although Professor Agostino Pierro, who led the surgical team, warned it was still early days.
Outside the hospital this morning, he said her chances of survival were around 50-50 but that she would face more surgery if she continued to improve.
'It's fair to say the operation done on Hope and Faith was one of the most complex and challenging we have ever faced,' the surgeon said.
He revealed both twins could have died had they not performed the emergency surgery because of a blockage in their shared intestine.
The separation surgery, which involved a team of 20 experts, only finished at around 7am this morning and 'went according to plan'.
'However, very sadly, after separation baby Hope's lungs proved too small to support her breathing and she died last night in the presence of her parents,' Prof Pierro said.
'They are clearly devastated by the loss of their daughter and we offer them our deepest condolences on their loss.'
He added: 'Baby Faith is stable. She requires support for her breathing but she is gradually improving. However, it is early days and complications can occur.'
His team were forced to perform the emergency separation because a blockage in the twins' shared intestine made Faith and Hope's health deteriorate rapidly.
They were born by Caesarean section last Wednesday 35 weeks into the pregnancy and had a combined weight of 10lb 8oz.
The couple, who have another daughter Carly, 18 months, had already picked out their daughters' names and called the smaller girl Hope because she was more fragile.
The birth went well and the girls were immediately placed under the care of Professor Pierro, who is head of paediatric surgery at Great Ormond Street, which is the leading centre in Europe for Siamese twins.
Doctors had initially been hopeful both would survive because they were joined from the breastbone to the top of the navel but had separate hearts.
However, they knew there was a risk emergency surgery would be needed because they shared a circulatory system as well as being joined at the liver and intestine.
They would normally have waited to operate until the twins were older and stronger to give them a greater chance of survival but the intestinal problems forced their hand.
Their parents gave permission for the surgery yesterday after the twins' health went downhill on Monday night and they were rushed into theatre last night.
There was no immediate comment from Mr and Mrs Williams today. Yesterday, they had thanked doctors for their efforts as the lengthy operation took place.
The couple had been advised to have an abortion when a scan revealed they were expecting Siamese - or conjoined - twins, which occur when the single egg from which identical twins develop fails to divide properly after conception.
It usually happens in women aged between 25 and 40 and the survival rate is somewhere between five and 25 per cent depending on where the children are joined.
The Williams were told that the babies were unlikely to survive and the pregnancy could be so dangerous Laura might never have any more children.
But as they revealed exclusively in the Mail on Sunday, they refused to have a termination on moral grounds and decided to see the pregnancy through.