The agreement also resolved some claims brought by US and California environmental regulators who had sought penalties for Volkswagen's violations of pollution laws.
"Final approval of the 2.0-liter TDI settlement is an important milestone in our journey to making things right in the United States and we appreciate the efforts of all parties involved in this process," Hinrich Woebcken, president and CEO of Volkswagen Group of America, said in a statement.
The settlement offers compensation to nearly a half-million owners of polluting Volkswagen and Audi diesel-powered vehicles in the US which the German automaker has admitted configuring to deceive emissions tests.
Owners of 2.0-liter diesel-powered cars will be eligible for buybacks, penalty-free lease terminations or free modifications to fix the vehicles' emissions. All participants will also receive cash payments.
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"The settlement adequately and fairly compensates class members," US District Judge Charles Breyer wrote in an opinion approving the deal. Breyer had granted preliminary approval for the settlement in July, praising the "enormous efforts" by all sides to reach an agreement and avoid a trial.
California will receive about $1.2 billion to mitigate environmental damage caused by the pollution, of which USD 381 million will fund projects to reduce smog, such as creating incentives for cleaner heavy-duty vehicles, according to the California Air Resources Board.
"Today is a landmark day, when this innovative settlement can be put into action, investing billions of dollars into public health protections to remedy these serious violations," Cynthia Giles, head of enforcement at the US Environmental Protection Agency, said in a statement.
The costs of responding to the scandal have been considerable. Volkswagen reported a 1.6 billion euro (USD 1.74 billion) net loss for 2015, its first more than two decades. The company has booked 18 billion euros to cover repairs, buy-backs and legal costs for the scandal, but commentators have said the final amount could be much higher.
Volkswagen still needs to conclude a separate settlement in the United States concerning about 80,000 Volkswagen, Audi and Porsche cars with 3.0 liter engines equipped with the emissions cheat devices.
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