The 46-year-old becomes the Magpies' third manager of the season, following the sackings of Kevin Nolan in August and Harry Kewell earlier this month.
Ardley saved the Dons from relegation to the National League on the final day of the 2012-13 season before taking them into League One via the playoffs three years later.
"Having enjoyed very productive discussions with Neal and following close consultation with the board and key staff, I am very confident we have selected the right man to deliver our short and long-term targets," Hardy told the official website.
"Our biggest priority is preserving our proud status as the world's oldest Football League club and in Neal we have someone who knows what it takes to survive.
"Neither he nor I are under any illusions it's going to be easy but we are both confident of achieving this very important aim.
"Beyond that, Neal has demonstrated an ability to win promotion from League Two and sustain a club at a higher level. I hope, in time, he will be able to deliver the same for us."
Ardley, who left Wimbledon earlier this month and will be assisted by Neil Cox, added: "I would like to say a huge thank you to the chairman for giving me the opportunity to manage such a magnificent club.
"There are few clubs who could have tempted me into such a quick return to management following what was obviously an emotional departure from Wimbledon, a club I love dearly.
"But the chance to manage Notts County is one I couldn't let pass and I will come back from my break fully refreshed and determined to drive us up the table."
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