"It's really a grievous problem, not one that can be avoided lightly," Budget Committee ranking member Jeff SessionsJefferson (Jeff) SessionsFBI opens tip line requesting information on Charlottesville rally Sessions rails against Chicago during visit to Miami DOJ warrant of Trump resistance site triggers alarm MORE (R-Ala.) said just before the Tuesday vote. "Just last August, we agreed to certain debt limits. And we, I believe, have a moral obligation to not mislead the people who elected us when we said we intend to stay by the limits on increasing debt."

On Monday, Sessions argued the bill would require the Treasury to repay $11.4 billion in money that the U.S. Postal Service (USPS) overpaid to its employee retirement system, and would let the USPS defer $23 billion in payments to its retiree health benefit plan. In each case, Sessions said those funds would have to be offset by new government debt, but should instead by offset by spending cuts.

Rep. Susan CollinsSusan CollinsOPINION: Congress should censure Trump for his unfit conduct No. 2 Senate Republican backs McConnell in Trump fight The fight to protect the Affordable Care Act isn’t over MORE (R-Maine), one of the sponsors of the bill, repeated her arguments from Monday that the Congressional Budget Office analysis saying the bill costs $34 billion is "misleading." She said that the funds involved come from the USPS, and that returning them would not amount to any new burden on taxpayers.

"There are no taxpayer dollars authorized by this bill or appropriated by this bill," Collins said.

After brief debate, the Senate voted 62-37 in favor of waiving the point of order, just narrowly meeting the 60-vote threshold. Nine Republicans voted with Democrats to waive the point of order: Roy BluntRoy Dean BluntGOP debates tax cuts vs. tax reform Five tough decisions for the GOP on healthcare Lacking White House plan, Senate focuses on infrastructure MORE (Mo.), Scott Brown (Mass.), Thad CochranWilliam (Thad) Thad CochranGOP senators ask Trump to hold off on Venezuelan oil sanctions Both sides of the aisle agree — telemedicine is the future Overnight Finance: GOP offers measure to repeal arbitration rule | Feds fine Exxon M for Russian sanctions violations | Senate panel sticks with 2017 funding levels for budget | Trump tax nominee advances | Trump unveils first reg agenda MORE (Miss.), Susan Collins (Maine), John HoevenJohn HoevenGOP senator criticizes EPA head's closed-door meeting in North Dakota Senate GOP eyes end to August session McCain absence adds to GOP agenda’s uncertainty MORE (N.D.), Jerry MoranJerry MoranRepublicans rebuke Trump over Charlottesville remarks GOP senator wants classified briefing on North Korea McConnell faces questions, but no test to his leadership MORE (Kansas), Lisa MurkowskiLisa MurkowskiFeds to sell 14 million barrels from oil reserve Immigration battlefield widens for Trump, GOP Trump barrage stuns McConnell and his allies MORE (Alaska), Pat RobertsPat RobertsNo. 2 Senate Republican backs McConnell in Trump fight Overnight Healthcare: McConnell warns Senate not to block repeal debate | Insurers knock Cruz proposal | WH tries to discredit CBO | Lawmakers propose .1B NIH funding boost Trump: I’ll be ‘very angry’ if Senate doesn’t pass ObamaCare repeal bill MORE (Kansas) and Olympia Snowe (Maine).

With the point of order laid aside, the Senate was expected to immediately consider up to 39 amendments to the postal bill. Senate Majority Leader Harry ReidHarry ReidOPINION | 5 ways Democrats can win back power in the states THE MEMO: Trump's base cheers attacks on McConnell It's time for McConnell to fight with Trump instead of against him MORE (D-Nev.) said he would take up to 10 minutes per vote, but also encouraged some to be held by voice vote to speed up the process.

— This story was updated at 4:18 p.m. to add vote results.