The Edinburgh businesswoman had been working on a sale and purchase agreement for her £2.5million deal, which takes the club out of administration.
The Foundation, which will gradually buy the club from Budge within five years, wrote on its Twitter account: "Four historic words for all Hearts supporters 'deal done, thank you'. More info to follow."
The club's administrators, BDO, soon confirmed Budge had assumed majority control "following the successful completion of a sale and purchase agreement in addition to the conclusion of many other administration issues".
Joint-administrator Bryan Jackson said: "BDO will continue behind the scenes for a few more weeks and it is planned will exit from office before the start of the 2014/2015 football season with Hearts being managed by the new board of directors for the remaining duration of the administration.
"This is a great day for Ann, for the club, and for the fans who have stood by their team throughout a difficult year.
"There is no question that this has been a tough period for all involved but, to their credit, the players, management and fans have remained steadfastly committed and loyal during a time which would have tested even the most ardent of supporters.
"I am very pleased for all involved that we have had a positive outcome to this administration. There have been times over the last year when it was not clear that we would get this result.
"There remain many challenges ahead for the club but it is clear that all involved remain resilient and determined to secure and maintain the great name of Heart of Midlothian for decades to come."
The sale and purchase agreement was the last hurdle after administrators BDO helped persuade Lithuanian-based major creditors and shareholders, Ukio Bankas and UBIG, to accept Budge's offer and sell their shares.
Hearts went into administration last June with debts of about £30million and the Foundation of Hearts fan group, which has 8,000 members, secured preferred bidder status. The fans will have representation on the new board, which Budge will chair, and will eventually assume control.
The decision could now bring some clarity to the future of manager Gary Locke, whose contract expires after Saturday's Scottish Premiership game at St Mirren Park.
Hearts were unable to overturn a 15-point penalty but Locke has led them to five wins from their last seven games.
It may still take some time for Hearts to formally exit administration because of issues such as Budge's legal obligation to make an offer to all shareholders.
However, she has previously expressed confidence that the process will be complete before the start of next season.