The changes will see the executive team reduced by almost a third, shrinking from 16 members to 11.
Ms Bulford, who is currently the BBC’s managing director of finance and operations, will take on additional responsibilities with the BBC’s marketing and audience teams.
The last deputy director-general was Mark Byford, who left the BBC in 2011.
Lord Hall said the new executive will lead the process of “creative renewal”, ensuring “the BBC remains the most creative force in the world”.
The BBC said it had already saved more than 40% on its senior manager pay bill, reducing the overall pay bill by £150 million a year.
With the top team reshaped and reduced in number, the corporation has said that each area will now look at the “immediate layers below” to identify “further headcount and post savings”.
As part of the restructure, the directors for Scotland, Wales and Northern Ireland have been cut from the core executive team.
Instead, the new position of director of nations and regions has been created. The role involves overseeing the directors in Scotland, Wales, Northern Ireland and the regional hubs in the North, Midlands and South West.
BBC Scotland director Ken MacQuarrie, BBC Cymru Wales director Rhodri Talfan Davies and BBC Northern Ireland director Peter Johnston will no longer sit on the executive team.
The BBC has said that the nations directors will continue to play a crucial role.
Others to be removed from the team are David Jordan, who represents Editorial Standards, and Philip Almond, from Marketing and Audiences.
The director of the BBC World Service, Fran Unsworth, will also no longer sit on the executive team.
The corporation has decided not to fill the position of director of BBC North vacated by Peter Salmon. Instead, director of BBC Children’s, Alice Webb, will lead the region on a day-to-day basis.
BBC One controller Charlotte Moore will remain in the role but will take on further responsibilities as director of content. She will oversee all the BBC’s TV channels, as well as iPlayer and BBC Sport.
The senior leadership restructure also sees a greater focus on education, with James Purnell, currently director of strategy and digital, becoming the new director of strategy and education.
He will work on delivering the BBC’s new Ideas Service, which will aim to partner the BBC with leading academic, arts and science institutions such as the British Museum, the Royal Society and the Royal Shakespeare Company.
Lord Hall said: “On a practical level, these changes are also about creating a simpler BBC with fewer layers and clear lines of accountability. Much has already been achieved, but this process must now be accelerated.
“The BBC also needs to reinvigorate its offer to the public on education - something I am passionate about. That’s why its importance is now recognised at the very top of the BBC.
“The UK is changing and changing fast. The BBC needs to recognise and address that. That’s why the work of improving our news and programming offer to the nations and regions is so important.”