The 27-year-old joins the Tynecastle side on a deal until 2021 after going almost two years without a club following his departure from Barnsley.
White came through the ranks at Leeds United and made over 100 appearances before his move to Oakwell.
A troublesome groin injury forced him from the game, but having finally recovered he is now working back to full fitness.
Manager Craig Levein told Hearts TV: “We’ve been in dialogue with Aidy for a year and feel that now is the right time to bring him in.
“He’s got a lot of work to do in order to reach the level we think he’s capable of but I’m confident he can fulfill his potential and prove to be a vital cog in the wheel as we look towards the future.”