The Sons had been without a manager since parting company with Gerry McCabe on 11 November, with Jim Clark taking over on a caretaker basis.
Chapman returns to the club where he ended his playing career and began to cut his teeth in coaching.
Sons chairman Ian MacFarlane said he was confident Chapman would boost the club's league standing.
Dumbarton are currently second bottom of Division Three with 16 points from 18 games.
MacFarlane said the new boss would be tasked with developing local talent with a view to adding them to the playing squad.
"Like all fans I've been frustrated with our season so far," MacFarlane told the club website.
"However, Jim's style of management will bring with it a change in performance and fortune in the weeks and months ahead.
"This appointment also re-enforces the club's commitment to youth development as our long-term future.
"Jim's wealth of qualifications and experience in this area will help us to develop new local talent to meet this goal."
Chapman's first game in charge of Dumbarton will be against former club Albion Rovers on 2 January.
Bookmark with:
What are these?