The acquisition is aimed at expanding BAT's business in Central Europe.
TDR is the main independent cigarette manufacturer in the region, with a market lead in Croatia and a strong presence in Bosnia and Serbia.
As part of the deal, BAT has said it is committed to keeping TDR's factory open in Croatia for at least five years.
BAT's chief executive, Nicandro Durante, described it as an "exciting acquisition".
The company said it hoped to benefit from a skilled work force, well established brands and strong existing relationships with distributors and retailers.
As smoking declines in many western markets, tobacco companies are increasingly looking for new markets or opportunities to consolidate in order to reduce costs and boost profits.
The acquisition is expected to be completed by October of this year, subject to approval from competition regulators and Adris Grupa's shareholders.
Adris Grupa's other business interests include tourism, insurance and food.
British American Tobacco (BAT) has agreed to buy cigarette firm TDR from Croatia's Adris Grupa for â‚¬550m (Â£395m).