The Obama administration has asked a federal appeals court to uphold a race-conscious admissions system at the University of Texas at Austin, aiming to stymie a lawsuit that conservatives hope will spur the Supreme Court to limit affirmative action at public colleges.

The Texas case tests a 2003 Supreme Court decision that upheld a race-conscious admissions system at the University of Michigan Law School. That ruling in Grutter v. Bollinger said the law school had "a compelling interest in attaining a diverse student body."...