The brave five-year-old had the operation last Friday at St James's Hospital in Leeds – and, incredibly, she is already on her feet.
Yesterday Bethany walked all the way from her bed on the liver unit to the play room down the corridor.
Mum Sam said Bethany had been so proud of her achievement, she couldn't wait to tell her dad Tony what she had done.
She said: "She couldn't make it back again but it was amazing. She was grinning from ear to ear."
The family got the call telling them that a liver was available on Friday morning. Bethany was rushed into hospital by ambulance at 8.30am.
Sam said: "It was a rush job, we had to go as quickly as we could, which Bethany thought was fantastic. We opened the window so she could listen to the sirens."
After a long operation, Bethany was out of theatre and onto the intensive care ward by 7pm.
Doctors found her new liver was functioning as it should do and she was transferred onto the liver ward on Sunday.
Messages of support have been flooding in to the family's website from all over the world.
And Sandra Partington – the wife of Dewsbury rector Kevin Partington, who had a life-saving kidney transplant more than 20 years ago – said she was very pleased for the family.
"This is absolutely wonderful news and a great thank you to those who had clearly talked about and supported the question of organ donation," she said.
"It underlines my own appeal for everyone to consider going on the donor register and also giving blood.
"Both Kevin and I are so, so pleased for the Salmon family.
"We will be praying for Bethany because it is still early days. I had hiccups with my new kidney and I just pray, I really do, that she has a smooth journey from here. We are both so thrilled."
To follow Bethany's progress on the family's website, go to http://www.caringbridge.org/visit/bethanysalmon.