The General Motors Corporation, the world's largest private provider of health care, won a federal judge's approval yesterday to proceed with $1 billion in cuts in the benefits of its retired workers.

The cuts will require more than 475,000 retirees and their dependents to pay part of their health insurance premiums for the first time. The annual cost will be as much as $752 a family.

Reducing G.M.'s health care bill is a central part of the effort by the chief executive, Rick Wagoner, to trim $7 billion in annual costs and restore profit after the automaker posted losses of $10.6 billion in 2005.

The ruling by Judge Robert H. Cleland of Federal District Court in Detroit came after his preliminary approval on Dec. 22. G.M. and its largest union, the United Automobile Workers, had agreed in October to help lower the company's health care spending.

The U.A.W. had sued G.M. over the automaker's threat in June to make unilateral cuts in retiree health benefits, claiming that it was unlawful. The October agreement resolved that dispute.

After Judge Cleland's preliminary ruling on the benefit cuts, G.M. notified its retirees and the attorneys general of all 50 states.

The judge had to wait at least 90 days after that notice to issue a final decision. About 20 G.M. retirees and dependents objected to the plan at a March 6 hearing, a G.M. spokeswoman, Sharon Baldwin, said.

G.M. has 1.1 million active and retired employees and dependents.

The U.A.W. and the Ford Motor Company agreed to a similar health care savings plan on Dec. 10.