Daphne Dearing, 62, only discovered last week she was an exact match for Susan Daunton, who has been ill for more than a decade.
Mrs Daunton, who emigrated in 1980, now has polycystic kidney disease.
The idea of a donation emerged last year after a consultant - not aware of the existence of Mrs Dearing - said her best hope was to "discover a twin".
Exact match
Mrs Dearing, a care worker, said: "It didn't come into my mind, and I just accepted the fact her kidney was failing.
"Then one day her consultant said: 'You could do with a twin', at which point she told him she had a real-life one in England.
"She started joking with me over the phone saying 'I'll have one of yours'.
"When I found out I was not just a good match but an exact match I was ecstatic. We were both so elated, even though we knew deep down that I would be suitable."
" I'm not concerned by the risks - you could get run over by a bus tomorrow"
She in now waiting to find out when she and her husband Malcolm are to fly to Australia.
It depends on space being free at hospital for the operation, which is expected to be in the next six months.
Mrs Dearing, from Patchway, will remain on the other side of the world for up to two months while she recovers.
She will pay for her own flights and is even expecting to fund part of the treatment as there is no National Health Service in Australia.
"There is a one in 3,000 chance of me dying and there is also the risk of her rejecting the organ," said Mrs Dearing.
"But that is unlikely as we are such a good match. I'm not concerned by the risks - you could get run over by a bus tomorrow."
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