WASHINGTON — The Senate has passed historic immigration legislation offering the hope of American citizenship to millions, while promising a military-style surge to secure the border.

The vote was 68-32, far more than the majority needed to send the measure to the House. Prospects there are not nearly as good and many conservatives are opposed.

Vice President Joe Biden presided, and senators cast their votes from their desks, both steps reserved for momentous votes.

The bill, a priority for President Barack Obama, would amount to the most sweeping changes in decades to the nation’s immigration laws.