The deal will formally take effect after Calico’s annual general meeting on 28 October.
Calico, which comprises four companies, three of which are registered charities, provides a range of housing and support services across north-west England.
The group employs about 350 people and its projected turnover this year is £38m.
Acorn, which will retain its identity as part of the Calico Group, has an annual turnover of more than £2m and has 75 employees.
Michael Birkett, chief executive of Calico, said that Acorn had approached Calico about the move.
"By welcoming Acorn into the Calico Group, we are continuing to grow the range of services that we provide to commissioners and their clients across the north," he said.
"As a group, we are able to provide property and a range of diverse support services for vulnerable people, and we strongly believe that this will give both value for money and quality for the organisations and people we work with."
Birkett said there would be "absolutely no redundancies".
Graham Elliott, chair of Acorn, has been appointed to the Calico board. Acorn’s chief executive, John Hopkins, will report to Birkett and will join the group’s executive team.
In a statement, Hopkins said: "Acorn was an existing partner of Calico – we had previously worked together on a supported housing scheme in Rossendale, Lancashire."We felt that it would be beneficial to have the formal support of a housing association if we were to continue to thrive. The support we will receive from being a part of the Calico Group will allow us to provide our services to even more people across a wide geographical area."