Graphico, which was founded in 1990 and employs around 70 people, specialises in digital and mobile marketing, and has a client list that includes First Great Western, Ann Summers and British Gas.
DMG, which is headed by former Euro RSCG London chairman Ben Langdon, said the acquisition was worth a maximum amount of £8.6m and was being funded by existing cash resources, loan notes and shares in the business.
DMG has also completed the acquisition of digital agency Hyperlaunch for a maximum consideration of £3.5m, and the company said it and Graphico formed part of its "buy and build" strategy for on and offline direct and database marketing businesses.
Hyperlaunch, which was founded in 2001, specialises in creative digital campaigns and has a client list that includes Universal Music, Atlantic Records and Warner Bros.
The deals follow the acquisition of digital media planning and buying agency Cheeze and Alphanumeric Group in January for around £24m in cash and shares.
Langdon, chief executive of DMG, said: "The businesses will bring new expertise and skills into the group and complement further our specialist digital marketing offering. We look forward to welcoming them into the group and working together to develop and create one of the UK's leading digital direct marketing groups."