The latest technique, called in-vitro maturation (IVM), involves taking tiny immature cells called oocytes and growing them into mature egg cells in a laboratory.
These egg cells can be implanted in 20 or 30 years' time. Because they are single egg cells, the risk of cancerous cells being implanted alongside them is eliminated.
Doctors from Oxford University and the Oxford Fertility Centre said the difficulty in such a young patient is that the immature egg cells are extremely small.
Tim Child, medical director of the Oxford Fertility Unit, which carried out the operation, said: "We use a laparoscopy to remove part of an ovary but before it's frozen we examine the tissue to find the most mature of the immature eggs, using a microscope to see if there are fluid-filled sacs or follicles which contain them.
"The bit of tissue has thousands of immature eggs but we can't see them because they are so small - what we are doing is looking for the ones which are slightly less immature.
"We then put them in a culture in an incubator and incubate them overnight."
The culture contains hormones and growth factors that encourage the cell to mature.
He said: "I think this is great on two fronts. Cancer treatment can be very successful but the drugs can completely damage the ovaries.
"This gives hope to young girls who could otherwise be sterilised by chemotherapy or radiotherapy. It's giving them the hope of one day having children.
"It's extremely exciting because it's two simultaneous approaches showing effect.
"What we want is to find a way to retrieve large numbers of immature eggs, the ones that are so small they can't even be seen under the microscope.
"That would be the holy grail of fertility preservation because the possibilities are limitless."
Mr Child said it was not known whether the eggs would result in pregnancy but he said he was "extremely hopeful".
He said: "It could be two or three decades before children given the treatment now want to start a family.
"I am extremely hopeful and excited to think where we could be in 20 or 30 years when these patients want to start a family."
Mr Child will present his research to the European Society for Human Health and Reproduction's conference in Helsinki next week.
Women are born with about two million oocytes, contained in tissue called follicles, but by puberty this number has fallen to around 400,000.
Usually, one follicle releases an oocyte that has matured into an egg during ovulation. Over a woman's reproductive lifetime, just 400 egg cells are released.
Consultant gynaecologist Stuart Lavery, of Hammersmith and Queen Charlotte's Hospital, said: "It is really amazing. What the Oxford team have done is to be able to find and mature eggs.
"It involves laparoscopic surgery, to remove a piece of the ovary. We know this can be done - its been done about 30 times now in adults.
"But the issue is if you remove pieces of ovarian tissue from a pre-pubertal girl there will be no eggs.
"What they did here using in-vitro maturation was to grow the egg in the lab, and then to freeze it.
"It's quite extraordinary but the real success will come when we know if its possible to successfully defrost and implant."
The world's first birth from a frozen ovary tissue took place in 2014.
A 28-year-old in Belgium gave birth after her tissue had been removed when she was 13, shortly before undergoing chemotherapy treatment.
