The clip was taken from Jones' 2GB radio programme on August 15, and was part of his original attack against Ardern.
In it, the audibly angry host can be heard saying: "This lightweight New Zealand Prime Minster Jacinda Ardern is challenging Scott Morrison over climate change. Now I hope Scott Morrison gets tough here with a few backhanders … hasn't got a clue this woman."
The comments received widespread condemnation, with former prime minster Malcolm Turnbull describing it as "appalling" and a "misogynistic rant", with Fijian Prime Minister Frank Bainimarama also slamming Mr Jones on Twitter.
Members of the public also voiced their outrage over the broadcaster's words, with many calling on advertisers to pull their sponsorship from his program and the network.
That pressure worked, with more than 100 companies withdrawing their advertising dollars, which the Sydney Morning Herald claimed cost parent company Macquarie Media more than A$1 million.
That mass exodus of corporate advertisers was largely due to concerted campaigns by social media activist groups such as Mad F**king Witches and Sleeping Giants Oz.
Now Macquarie Media has apologised for Jones' rant, reportedly sending a letter to advertisers today pledging a review of Jones' Breakfast Show on 2GB and 4BC.
"Through this incident, we have experienced the ability of offended groups to greatly amplify their complaints and to actively disrupt you, our clients and your staff, who have done no more than seek to engage with the audience who listens to us," company chairman Russell Tate wrote.
"Of course, we have seen valued commercial partners withdraw from Alan's programme, but the fact is we got it wrong in the first place and we must now do everything possible to ensure that doesn't happen again."
Despite Jones apologising, he was told if he made similar comments, his contract would be terminated.
- additional reporting news.com.au