WASHINGTON—Senate Republicans adopted a budget for the next fiscal year, clearing a critical hurdle in the GOP push to overhaul the tax code.

The Senate’s late Thursday passage of the budget blueprint, in a 51-49 vote primarily along party lines, helps unlock a procedure that Republicans plan to use to rewrite the tax code with just GOP votes. It also allows the tax bill to lower projected revenue by up to $1.5 trillion over a decade. Sen. Rand Paul of Kentucky was the lone Republican to vote against the budget.

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