A Swedish woman who gave birth after a womb transplant has brushed off the stresses of caring for a newborn as she marvels at her miracle baby.

The parents of the first baby born to a woman who had a womb transplant say they hope they can be an inspiration to others struggling with infertility.

After what they describe as a rollercoaster of a journey, the unnamed Swedish couple finally became parents last month, when the mother gave birth to a healthy but premature baby boy. For the couple, making history was an afterthought.

"Yes, we're the first to do this, but that's not the important thing," she said during an interview at their home, in an undisclosed part of Sweden. "I want people to know that what they think is impossible can happen."

To mark their baby's birth as a victory over their difficult journey to parenthood, they named him Vincent, meaning "to conquer".

Cradling her sleeping young son, she said she still could not believe she was finally a mother, after being told at 15 that she had no womb and would never carry her own children. Now 36, she was one of nine women to receive a transplanted womb last year in a ground-breaking trial led by Dr Mats Brannstrom, a professor of gynaecology and obstetrics at the University of Gothenburg and Stockholm IVF.

For the new mother, the normal stresses of caring for a newborn seem less important after years of thinking she would never have a child.

"We've been through so many things I couldn't control but now that this little fantastic boy is here, I just feel relieved," she said.

Her husband said they will be forever grateful to the 61-year-old woman who donated her uterus, the mother of one of his best friends. She made the offer after hearing about the difficulties the young couple was having in starting a family.

"What she did for us was so amazing and selfless that the words 'thank you' don't seem like enough," said the father, adding she is also the godmother to their son.

These days, the new parents are busy marvelling at their baby's expressive face and remarkably calm nature.

"He doesn't really scream, but he makes these funny little sounds," the mother said, comparing him to a little kitten.

She and her husband said they haven't quite figured out how to tell their son how he made medical history, once he's older.

"We will show him all the articles that were written and tell him everything we went through to get him," she said. "Maybe he will be inspired to become a doctor."

Copyright AAP 2014.