Under the five-year contract, worth £120m, Circle will manage services that were previously handled by more than 20 different providers.
This is the first time that a major integrated NHS service has been led by a single private sector contractor.
Circle will work in partnership with Pennine MSK, which developed the original integrated service in Oldham, and with local charities and clinicians.
Biggleswade GP Andy Edwards said: “I look forward to working with Circle and their partners in this exciting and innovative new venture and to seeing the benefits this new model will deliver in improved care for local patients and their families.
“As a local GP, I want my patients to have access to high-quality, modern, integrated services as close to home as possible.
“This plan offers our population now and in the future this opportunity and by working together with patients, national and local experts we can see real improvements very quickly.”
In February 2012, Circle became the first private firm to take over an NHS acute trust; however, its management of Hinchingbrooke Hospital in Cambridgeshire has struggled financially, and in December 2012 Circle CEO Ali Parsa resigned.
“This new integrated approach is essential for the NHS to meet the growing challenge of delivering high-quality care at an affordable cost,” said new Circle CEO Steve Melton.
“We are delighted to have this opportunity to work with local doctors and other healthcare professionals to join up musculoskeletal service in Bedfordshire and help improve care for patients.”