"The boards of directors of Deutsche Bahn and Arriva are pleased to announce that they have reached agreement on the terms of a recommended cash offer," the German group said in a statement published by the London Stock Exchange.
"Under the terms of the acquisition, Arriva shareholders will receive 775 pence in cash for each Arriva share they hold, valuing the entire issued and to be issued share capital of Arriva at approximately €1.585 billion."
Arriva ranks among the top British transport companies with a presence in 12 European countries where it operates bus and metro systems.
Deutsche Bahn chief executive Rudiger Grube on Thursday said that the deal for Arriva was part of his group's strategy to take advantage of increasingly liberalised European transport markets.
The German group is also expanding as private sector competition erodes its home market share. Deutsche Bahn already owns Chiltern Railways in Britain.
Grube said in the company's statement: "Arriva's activities will strengthen Deutsche Bahn's strategic positioning in Europe, principally through Arriva's successful targeting of Europe's increasingly liberalised and fast growing transport markets which are of strategic interest to Deutsche Bahn.
"Arriva will give Deutsche Bahn the platform to expand in Europe and enhance its position as one of Europe's leading passenger transport groups," he added.
Richard Broadbent, chairman of Arriva, said Deutsche Bahn's offer "fully reflects" the value of the British group.