San Diego ? California voters will be allowed to cast a ballot for a potential successor to Gov. Gray Davis even if they do not vote on whether he should be removed from office, a federal judge ruled Tuesday in striking down part of California’s recall law.

The ruling will not stop the Oct. 7 recall election to remove Davis. Still, the governor’s difficulties lessened somewhat Tuesday when a compromise budget intended to cover the state government’s record deficit and break a political stalemate passed the state Assembly.

Meanwhile, a Republican consultant close to Arnold Schwarzenegger said it was extremely unlikely that the actor, a Republican activist, would run.

“I think he’s not going to run,” said the consultant, who spoke on condition of anonymity.

Tuesday’s federal court ruling will change how the Oct. 7 recall election is carried out. Under state law, voters could choose “yes” or “no” on whether Davis should be recalled. And only voters who cast a “yes” or a “no” could then choose a potential successor from a list on the same ballot.

In his ruling Tuesday, U.S. District Judge Barry Moskowitz shot down the part of the law that said voters who did not answer the first question would not have their votes on the second question tallied.

“What is at stake is the right of a voter to decide who shall succeed the governor, if recalled. Every voter, whether they voted for or against that recall, has a paramount interest in choosing the person who will govern them,” Moskowitz said.

Terry Carbaugh, a spokeswoman for Secretary of State Kevin Shelley, said there were no plans to appeal.