"It's really a grievous problem, not one that can be avoided lightly," Budget Committee ranking member Jeff SessionsJefferson (Jeff) Beauregard SessionsIntel leaders: Collusion still open part of investigation Republicans jockey for position on immigration Biden to Alabama: No more extremist senators 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 CollinsGun proposal picks up GOP support Giffords, Scalise highlight party differences on guns Agricultural trade demands investment in MAP and FMD 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 BluntThe Hill's Whip List: Republicans try again on ObamaCare repeal Another health funding cliff puts care for millions at risk Top Senate Dem: We're going forward with understanding we can work with White House on DACA MORE (Mo.), Scott Brown (Mass.), Thad CochranWilliam (Thad) Thad CochranWhite House requests B for disaster relief GOP establishment doubts Bannon’s primary powers Whatever you think the Alabama special election means, you’re probably wrong MORE (Miss.), Susan Collins (Maine), John HoevenJohn Henry HoevenThe Hill's Whip List: Republicans try again on ObamaCare repeal Air Force One is Trump’s new boardroom Overnight Finance: Trump strikes debt, spending deal with Dems | Deal shocks GOP | Fed’s No. 2 to resign | Trump keeps tax squeeze on red state Dems | House aims to pass budget next week MORE (N.D.), Jerry MoranGerald (Jerry) MoranTIMELINE: The GOP's failed effort to repeal ObamaCare The Hill's Whip List: Republicans try again on ObamaCare repeal IT modernization measure included in Senate-approved defense policy bill MORE (Kansas), Lisa MurkowskiLisa Ann MurkowskiRepublicans jockey for position on immigration GOP senator knocks Trump: 'Not a fan of governing by tweet' How the effort to replace ObamaCare failed MORE (Alaska), Pat RobertsCharles (Pat) Patrick RobertsThe Hill's Whip List: Republicans try again on ObamaCare repeal No. 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 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 ReidChris Murphy’s profile rises with gun tragedies Republicans are headed for a disappointing end to their year in power Obama's HHS secretary could testify in Menendez trial 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.