The deal, for an undisclosed sum, will see Stirling Park's 130 employees, including 23 sheriff officers, move to Capita.
"All employees will remain with the business, as far as I know," a Capita spokeswoman said.
Kilmarnock-headquartered Stirling Park is one of the largest sheriff officer businesses in the country, responsible for collecting around £350 million of debt a year.
It operates out of eight premises in Scotland.
Founded in 1924 by the grandfather of Jonathan and Adam Lewis, two of the five partners in the business who have sold it to Capita, Stirling Park started as Jack Lewis later becoming Jack Lewis and Sons.
In the 1980s the firm changed its name to Stirling Park and was bought by Swedish-owned Intrum Justitia in 2002.
A management buyout in 2009 saw it revert to Scottish ownership.
It provides a debt collection and sheriff officer service to 11 of Scotland's 32 local authorities and works for utility companies and financial services firms.
It also acts for the majority of Scotland's law firms as sheriff officers.
Neil Smith, director in Capita's debt management business, said: "The acquisition of Stirling Park will provide Capita with the scale and focus to grow its debt management business by expanding into the Scottish market and providing a complementary service to our new high court service in England and Wales.
Capita has bought a number of business over the last year to build a debt and legal services arm. This includes iQor (now Akinika), acquired for £42 million last year, Optima Legal and analytics firm Euristix.
The group said it can now provide support and services from first notification of debt to collections and enforcement.
Capita already employs around 5,000 people in Scotland - up from 500 in 2006 - in areas including Information technology, financial services, education, property, utilities and telecoms.
The company said in October that it expects to boost its headcount north of the Border by around 2,000 in the next two years.
At the moment the majority of Capita's work in Scotland is in the private sector with financial services and it has a large call centre in Stirling.