By putting the new liver cells into mice, they showed they were fully functioning.
"We have developed new systems to target genes and ... correct ... defects in patient cells," said Allan Bradley, director of the Sanger Institute.
At a briefing about the work, Bradley said the technique -- the first success of its kind -- leaves behind no trace of the genetic manipulation, except for the gene correction.
"These are early steps, but if this technology can be taken into treatment, it will offer great possible benefits for patients," he added.
Stem cells are the body's master cells, the source for all other cells. Scientists say they could transform medicine, providing treatments for blindness, spinal cord and other severe injuries, and new cells for damaged organs.
Research is focused on two main forms -- embryonic stem cells, which are harvested from embryos, and reprogrammed cells, also known as induced pluripotent stem cells or iPS cells, which are reprogrammed from ordinary skin or blood cells.
When they were first discovered in 2006, iPS cells looked like a perfect solution to the ethical debate over the use of embryonic stem cells because they are made in a lab from ordinary skin or blood cells.
Embryonic stem cells are usually harvested from leftover embryos at fertility clinics and their use is opposed by many religious groups.
But in recent years, concerns have been raised that iPS cells may not be as "clean" or as capable as embryonic cells.
Last year, a group led by Robert Lanza, of the U.S. firm Advanced Cell Technology, compared batches of iPS cells with embryonic stem cells and noticed the iPS cells died more quickly and were much less able to grow and expand.
In Wednesday's study, published in the journal Nature, the British team took skin cells from a patient with a mutation in a gene called alpha1-antitrypsin, which is responsible for making a protein that protects against inflammation.
People with mutant alpha1-antitrypsin are not able to release the protein properly from the liver, so it becomes trapped there and eventually leads to liver cirrhosis and lung emphysema.
This is one of the most common inherited liver and lung disorders and affects about one in 2,000 people of North European origin, the researchers said.
Having harvested the skin cells, the scientists reprogrammed them back into stem cells and then used a type of "molecular scissor" technique known as a zinc finger nuclease to snip the cells' genome at precisely the right place and insert a correct version of the gene using a DNA transporter called piggyBac.
The leftover piggyBac sequences were then removed from the cells, cleaning them up and allowing them to be converted into liver cells without any trace of residual DNA damage at the site of the genetic correction.
"We then turned those cells into human liver cells and put them in a mouse and showed that they were viable," David Lomas, a Cambridge professor of respiratory biology who also worked on the team, told reporters at the briefing.
Ludovic Vallier, also from Cambridge University, said the results were a first step toward personalised cell therapy for genetic liver disorders. "We still have major challenges to overcome...but we now have the tools necessary," he said.
The researchers said it could be another five to 10 years before full clinical trials of the technique could be run using patients with liver disease. But if they succeed, liver transplants -- costly and complicated procedures where patients need a lifetime of drugs to ensure the new organ is not rejected -- could become a thing of the past.
"If we can use a patient's own skins cells to produce liver cells that we can put back into the patient, we may prevent the future need for transplantation," said Lomas.