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Marsh, 19, the younger brother of top-order batsman Shaun, was selected to take Lee's place in the one-day squad, with chairman of selectors Andrew Hilditch describing him as an upcoming talent.
"Mitchell is a very exciting young all-rounder who we consider has the ability to have a significant impact at international level," Hilditch said.
"He played extremely well during the recent Australia A tour of Zimbabwe and is a pace bowler who has the ability to bat in the top six, providing great balance and flexibility to the squad."
Marsh is already in South Africa as part of Australia's Twenty20 squad and was due to fly home after this week's T20 games.
Lee was ruled out of the South Africa tour earlier this week with appendicitis and underwent surgery in Cape Town on Monday night.
Hilditch said the depth of bowling talent in the one-day squad, including Doug Bollinger, Mitchell Johnson, James Pattinson and Patrick Cummins, meant selectors "did not consider it necessary to replace Brett with a specialist fast bowler."
Marsh captained Australia's under-19s in their 2010 victory at the cricket world cup in New Zealand, and the current tour of South Africa was his first first-class international call-up.
Hilditch said Marsh was "very close to selection in the one-day squad when it was initially picked.
"Brett’s unfortunate illness has provided a great opportunity for him to gain further experience at the international level," he said.
