Eight-year-old Ann Stephen, from Hengrove, had previously said that she wants to meet the pontiff to see if he is able to make her better.
Ann was diagonised with a form of cancer, acute lymphoblastic leukaemia, in October 2013.
Ann's mum Reena told The Bristol Post: "When we were told Ann's diagnosis we were utterly shocked. It was so distressing we didn't know how to react."
Ann's mother said: "We couldn't believe our eyes when we read what Ann's wish was - to meet His Holiness Pope Francis.
"We're a Catholic family and the children attend Catholic school but we were surprised and confused about where Ann's idea had come from."
After Ann was asked why she had chosen this wish, she told her parents that she had seen Pope Francis bless children on television on his USA trip and she had hoped he might be able to make her better.
Pope Francis kisses a child during a trip to Mexico
The girl's dream was able to come true after Starlight, a charity caring for children with terminal illnesses, arranged the trip.
Her father Siby said: "Of course our immediate reaction was to think that this type of wish would be impossible to grant and we didn't want to get Ann's hopes up.
"We explained to her that it may or may not happen but that her Wish Granter, Amy, would do her very best."
Pope Francis with a child on his Mexico visit
Ann's parents and sister and brother all travelled to Rome with her and met the Pope at St Peter's in the Vatican.
Siby added: "Ann couldn't stop smiling but she was quite quiet, I think we were all a bit overwhelmed, it felt like a dream."
Ann was then given a blessing by Pope Francis, and signed photos and rosaries.
Ann said of her experience: "It was incredible. Thank you Starlight for making it happen, I'll never forget it."
