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How the General Assembly will fund Mr. McDonnell’s budget priorities remains unclear, considering he and the Republican-controlled House have consistently rejected the idea of tax increases.

Shortly after taking office, Mr. McDonnell and the legislature closed a $4.2 billion budget gap without an increase in taxes, in part by slashing funds for health care, education and public safety.

“Even in tough budget years, you always have to think about the future,” Mr. McDonnell told state employees at a town-hall meeting last week. “Not everything in the budget is going to be cause for celebration.”