Marsh became the latest Australian player to pull out midway into the IPL season due to injury.
Amla confirmed the development on Twitter that he would be joining the bottom-placed team in the IPL league table.
"Excited to be with @lionsdenkixp for the remainder of the @ipl. Looking forward to it. Thanks for the support peeps," Amla said.
The 33-year-old's signing comes after Marsh failed to shake off a back injury which he reportedly picked up in a game against the Kolkata Knight Riders on April 19.
Skipper Steve Smith, bowler John Hastings and Marsh's brother Mitchell are the other Australian players to have gone home owing to injuries.
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