DMG's initial payment is £6m cash, plus £3.5m in shares, with a further tranche, currently valued at £1m, on offer subject to business performance.
Cheeze, which was previously owned by founders Katherine Jerman and Jamie Riddell, is mainly a digital direct marketing agency with offices in Ipswich, London and Leeds.
Its core services are planning and buying for digital media campaigns. It also offers search engine marketing, customer relationship marketing and digital direct marketing consulting.
Unaudited financial reports for last year show Cheeze had turnover of £10.5m, a profit before tax of £768,000 and net assets of £950,000.
DMG chief executive Ben Langdon said: "We have defined our space as the digital direct model."
He added that Cheeze "naturally complements" its digital interactive agency Inbox and direct marketing specialists Dig for Fire.
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