The mother conceived naturally and gave birth to a baby boy two weeks ago.
Edinburgh University scientists are freezing tissue from the reproductive organs of boys and girls as young as one, which can be re-implanted once they reach adulthood.
The 33-year-old mother had a section of her ovary removed 11 years ago.
She wishes to remain anonymous.
Following her chemotherapy, doctors re-implanted the tissue last year.
Researchers said the new service was open to NHS patients.
The new mother said: "That the re-implanted tissue took so quickly, came as a really wonderful surprise.
"I'm incredibly appreciative of my oncologist's foresight in sending me for the consultation with the fertility team.
"I had one small surgical procedure before I began my second round of chemotherapy and now, 10 years on, my husband and I have been able to have a family."
She added: "We never thought it would be possible and we are just astonished and overjoyed. We are extremely grateful to all the people involved in this process.
"When you're going through cancer treatment it can be hard to think about the future, but I do think this will offer hope to others that they could one day have a family."
Prof Richard Anderson, of Edinburgh University, said: "The storage of ovarian tissue to allow restoration of fertility after cancer treatment in girls and young women was pioneered in Edinburgh over 20 years ago, and it is wonderful to see it come to fruition.
"This gives real hope to girls and young women facing treatment that may cause them to become infertile, and shows how some medical advances can take a long time to show their benefits."
The research has been funded by the Wellcome Trust, Children with Cancer, the European Union and the Medical Research Council.
It has involved close collaboration with the Scottish National Blood Transfusion Service.

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12 September 2013 Last updated at 22:08 BST
In a BBC interview with Juliana Liu at the World Economic Forum's conference in Dalian, China, Mr McGregor, a former journalist who has lived in the country for more than 25 years, said China's behaviour was "very worrisome" for foreign companies. "They don't know what's hitting them right now", he said.
The government is resorting to its traditional "toolbox" of coercive methods, he argues, including "shaming and ordering people to confess that they've done wrong so that your penalties can be minimised".
At the same time as offering entrepreneurs more opportunities, he argues, the Chinese government is saying: "Don't mess with us".
"They're just treating foreign companies the way they've treated their own for many years, and this is the way the Party does things", he says. "These are the tools it knows."
"What may be going on is they're telling foreign companies and they're telling private companies here: Behave yourself; remember we're the Party, we're in charge."
His comments come against the backdrop of a Chinese police investigation into alleged corruption by GlaxoSmithKline (GSK), the British pharmaceutical company accused of using travel agencies to channel up to ??320m of bribes to doctors and officials.
GSK has denied any corporate involvement in the alleged wrongdoing, whilst admitting that certain individuals may have behaved inappropriately.

South Wales Fire and Rescue Service was called to the Butetown tunnel just after 01:00 BST.
Two crews from Cardiff Central, two from Ely and a crew from Penarth went to the scene.
South Wales Police said the A4232 westbound tunnel remained closed after parts of it were damaged in the crash.
A spokesman for the Welsh Ambulance Service said the woman had been taken to Cardiff's University Hospital of Wales with unknown injuries.
A cancer patient from Edinburgh has become the first UK woman to give birth following a transplant of her frozen ovary tissue.
[NEXT_CONCEPT]
Jim McGregor, Greater China chairman of communications consultancy APCO Worldwide, has given a critical assessment of the way China's government is treating foreign companies as part of its anti-corruption drive.
[NEXT_CONCEPT]
A woman has been taken to hospital following a crash in Cardiff in the early hours of Monday morning.