The charity would not reveal how much it paid for Prospects, which had revenue of £80m in the 2016/17 financial year and employs 1,400 people.
"The price was a fair and proportionate one for both parties in line with the due diligence and valuation that was conducted," the Shaw Trust said in a statement.
Prospects was formed in 1995 after the privatisation of a group of south London careers services. It now operates across the UK in various sectors, including careers, children's services and employment and skills.
Prospects joins the Shaw Education Trust, Ixion, Forth Sector and the Disabled Living Foundation as part of the newly formed Shaw Trust Group.
A spokeswoman said Prospects would remain as a for-profit subsidiary "in the immediate future" and there were no plans to make immediate changes to the workforce.
"Both Prospects and Shaw Trust will continue to deliver their prime contracts with the Education and Skills Funding Agency, the Department for Work & Pensions, the Welsh government and many local authorities, with contractual arrangements being unaffected by the merger," she added.
A group leadership team has been formed to integrate Prospects into the trust. Chaired by Roy O'Shaughnessy, chief executive of the trust, it also includes the trust's chief financial officer and the chief executives of Prospects, the Shaw Education Trust and Ixion.
An integration director is also being appointed to the team – they will be responsible for decisions that affect more than one company.
"The team will be reviewing the shape and structure of the group to enable it to effectively achieve its strategy, but there are no current plans to convert Prospects into a charity," the spokeswoman said.
O’Shaughnessy said in a statement: "I am delighted that Prospects has joined our group. We deliver many complementary services and now, together, we can offer an even greater range of support to our customers."
"This marks a very special day on our journey to transform the lives of one million young people and adults each year by 2022."
Nick Bell, chief executive of Prospects, said in a statement: "We have a very similar mission and values to Shaw Trust, and the merger of our organisations will enable us to create new opportunities and inspire more young people and adults a year to develop their potential."