"It's really a grievous problem, not one that can be avoided lightly," Budget Committee ranking member Jeff SessionsJefferson (Jeff) Beauregard SessionsSessions responds to Nazi comparisons: 'They were keeping the Jews from leaving' Laura Ingraham: Migrant child detention centers 'essentially summer camps' Senate chaplain offers prayer 'as children are being separated from their parents' 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 Margaret CollinsTrump plan to claw back billion in spending in peril Romney backs Laura Bush on border: 'We need a more compassionate answer' Amnesty International rips family separation policy: 'This is nothing short of torture' 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 BluntErnst, Fischer to square off for leadership post Facebook gives 500 pages of answers to lawmakers' data privacy questions Community development impact remains clear with NMTC post-tax reform MORE (Mo.), Scott Brown (Mass.), Thad CochranWilliam (Thad) Thad CochranTodd Young in talks about chairing Senate GOP campaign arm US farming cannot afford to continue to fall behind Mississippi Democrat drops Senate bid MORE (Miss.), Susan Collins (Maine), John HoevenJohn Henry HoevenGOP senators want NAFTA deal from Trump by Labor Day Senate GOP urges Trump administration to work closely with Congress on NAFTA GOP lawmakers want Trump to stop bashing Congress MORE (N.D.), Jerry MoranGerald (Jerry) MoranFormer USA Gymnastics CEO pleads Fifth at hearing GOP, Trump at odds on pardon power Lawmakers request meeting with Amtrak CEO over funding for route MORE (Kansas), Lisa MurkowskiLisa Ann MurkowskiTrump plan to claw back billion in spending in peril McCain calls on Trump to rescind family separation policy: It's 'an affront to the decency of the American people' GOP senator calls on Trump to end 'cruel' family separation policy MORE (Alaska), Pat RobertsCharles (Pat) Patrick RobertsTrump tightens grip on GOP Overnight Finance: Senators introduce bill to curb Trump's tariff authority | McConnell calls it 'exercise in futility' | Kudlow warns WTO won't dictate policy | Mulvaney feud with consumer advocates deepens Senators introduce bill to curb Trump's tariff authority 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 Mason ReidAmendments fuel resentments within Senate GOP Donald Trump is delivering on his promises and voters are noticing Danny Tarkanian wins Nevada GOP congressional primary 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.