It will screen at 9pm tonight, following a formal defamation settlement that includes paying all legal costs and some damages to Kenny, according to News Corp.
The move will also mean the Chaser team cannot republish the material or make public statements that detract from the settlement.
The sketch which aired last year on ABC1 resulted in a lengthy and embarrassing dispute for the broadcaster, leading to a belated apology by managing director Mark Scott.
“At the time of broadcast, I described the skit as tasteless and undergraduate, but noted that it raised questions about the nature of satire and the boundaries of free speech afforded to satirists, comedians and cartoonists,” he said in April.
“Notwithstanding any ACMA finding however, I have come to the view with the Director of Television that the ABC should not have put the skit to air.
“As a consequence, I would like to apologise to Mr Kenny for the ABC having put the skit to air, his depiction in the skit and because it was triggered by his criticism of the ABC. I am sorry for the distress this incident has caused him and his family.”
Kenny has previously indicated he wanted an apology rather than legal proceedings or settling out of court.
Last month ABC sent an ‘Open Letter’ to Kenny’s lawyers and media offering not to seek legal costs if it wins the case -an offer rejected by Kenny.