The 5,500 people in Britain waiting for a donor kidney, including more than 500 in Scotland, will be encouraged by the research.
A team of scientists in Israel said they had transplanted human and pig stem cells, destined to become kidney cells, into the mice.
They found both human and pig tissue grew into perfect, mice-size kidneys.
The team, headed by Professor Yair Reisner, of the Weizmann Institute of Science in Rehovot, said the human and pig kidneys worked correctly and produced urine as normal.
Significantly, the blood supply within the new organs was provided by mouse blood vessels, rather than donor blood vessels, which greatly lowers the risk of rejection.
The findings suggest that human or pig foetal tissue might take on the shape and function of a healthy kidney if transplanted into humans as well.
Pig tissue, rather than pig organs, is not expected to cause the type of rejection common in other cross species transplants.
The scientists hope this will mean pig stem cells could provide the material for growing new kidneys for people needing a replacement organ.
Asians are estimated to account for about a quarter of those on the UK waiting list, more than half of them in London and the West Midlands. This is because they are three times more likely to suffer kidney failure than white people, yet few Asians are organ donors.
Only about one in eight people in Scotland have signed up to the NHS organ donor scheme, and doctors have predicted demand will rise sharply over the next ten years as the population ages and medical advances increase the number of suitable recipients.
Britain’s first live kidney donor transplant took place in Edinburgh 40 years ago.
Kidneys filter more than 400 pints of fluid in the body for poisons every day, and produce some three pints of urine.
When the two kidneys are not working properly, harmful waste, including salts and fluids, build up, causing high blood pressure.
Among the key findings in the Israeli study, which was published in the journal Nature Medicine, is the pinpointing of the stage of embryonic development at which stem cells have the best chance of forming well-functioning kidneys with minimal risk for immune rejection. It suggested that seven to eight-week-old tissue in humans and four-week-old pig tissue offers the best window of opportunity for transplantation.
The research found that if the tissue was taken at an earlier stage, it could include unwanted non-kidney structures such as bone, cartilage and muscle.
However, they also discovered that if the tissue was taken later, the risk of rejection by the immune system was substantial.
And the scientists checked how human cells in the immune system might react to the human and pig tissue once it had been grown in the mice.
They injected such human "fighter cells", called lymphocytes, into mice lacking immune systems.
The researchers found that so long as the kidney tissue was transplanted within the right time range, the lymphocytes did not attack the new kidneys, even though they originated from two different donors.
The team said the research was in a pre-clinical study stage.
However, they added that if all went well, a treatment could follow within a few years.
They stated: "Kidney transplantation has been one of the major medical advances of the past 30 years. However, tissue availability remains a major obstacle.
"Our data pinpoint a window of human and pig kidney organogenesis [development] that may be optimal for transplantation in humans."
In September, American researchers said they had managed to grow teeth in rats, which suggested the existence of dental stem cells.
They said there was no reason why the technique used in rodents would not work in humans.
A month earlier, a British research team from King’s College, London, said it had become the first in the UK to grow human embryonic stem cells.
Also that month, the team that created Dolly the Sheep announced they had cloned pigs that were likely to prove suitable for supplying replacement organs such as kidneys and hearts.
A US subsidiary of the Midlothian-based PPL Therapeutics said the four pigs had been born without either copy of a gene which prompts the human immune system to reject pig tissue.
Pig organs are of similar size and power output to those of humans.
However, scientists do not yet know whether the risk of virus transmission from pigs to humans when transplanting organs can be overcome.
Heart valves from animals are already widely used for human transplants, but there is no viral transmission risk because they are not living tissue.
MICE have been biologists’ best friends for more than a century because they have a very similar genetic composition to humans - as this month’s sequencing of the mouse genome confirmed.
Some 25 million mice are used in laboratories across the world in research on everything from breast cancer and heart disease to malaria and osteoporosis.
They are the guidebooks used by scientists to help understand DNA in humans.
Both mice and humans have about 30,000 genes, with humans having only 300 not shared with the rodents.
They share a common ancestor, a shrew-like creature which lived at the time of the dinosaurs some 100 million years ago.
Comparison of the mouse genome with its human equivalent has revealed more than 1,200 previously undiscovered human genes.
Mice have been used to research the spectrum of human disease, since the mouse version of many ailments produces symptoms similar to those in people.
They provide a vital initial test bed for potential treatments to ensure they are safe to be tried on humans.
The mouse genome itself has already proved useful in studies such as on Down’s syndrome, diabetes and schizophrenia.
The genome is also expected to unlock the potential for using mice in research on epilepsy, obesity, colon cancer and hypertension.
Scientists at Dundee University recently used the mouse genome to research potential new treatments for arthritis.