The high-profile businessman, who replaces Jan du Plessis, will step down from his position as chairman of online fashion retailer Asos as he takes on the BT role from December 1, following a one-month transition with his predecessor.
He will remain chairman of Premier Inn owner Whitbread but will also step down as a non-executive of Sony, to avoid risks of so-called "overboarding" where a director can be accused of taking on too many boardroom roles.
Crozier, who was born on the Isle of Bute, called the appointment, which comes with a £700,000 a year salary, "an honour".
He said: "BT is a hugely important company, with a critical role to play in building the digital networks and services to support the UK's future.
"I look forward to working with the board, [chief executive] Philip [Jansen] and his executive team to create value for all our stakeholders."
Outgoing chair du Plessis said: "Chairing the board of BT over the past four years has been a tremendous privilege.
"BT is a truly unique and special business and what this company does really matters to so many people."
Jansen said: "I would like to thank Jan for his leadership over the last four years. He has overseen the achievement of significant milestones and the recent improvement in BT's fortunes and his careful stewardship has left the business in a better, stronger position.”
Iain Conn, BT's senior independent director, added: "After a thorough and comprehensive process to ensure we identified the very best candidate to lead BT, Adam is the unanimous choice of the board. He has significant experience in leading public company boards, developing teams and managing stakeholders and brings a strong transformational and operational track record in large-scale executive roles."
Prior to becoming a non-executive across several companies, Crozier shot to fame as chief executive of the FA in 2000, where he was responsible for appointing the first foreign manager of the England team.
He subsequently went on to run Royal Mail, turning it profitable and preparing the ground work for a stock market listing three years after his departure.
Crozier then turned his hand to running a media empire, when he spent seven years with ITV.
BT was forced to play down reports earlier this year suggesting the chief executive had fallen out with du Plessis over the direction of BT's turnaround, which includes spending huge sums on speeding up its broadband rollout.
Crozier will be expected to help oversee the £12 billion spending plans on full-fibre broadband to 20 million homes in the UK no later than the end of the decade, following the publication of new rules by regulator Ofcom in March this year.
He must also maintain good relations with the regulator - the same watchdog responsible for overseeing ITV - and is likely to offer his expertise on BT's TV services.
The new chairman will also need to manage the expectations of BT's new biggest shareholder, the billionaire owner of Sotheby's auction house, Patrick Drahi, who bought a 12.1 per cent stake in June.