The bill breezed through its second-reading stage, and every amendment from Coalition senators was defeated in a debate that lasted late into the night.

A proposal allowing civil celebrants to refuse to perform gay weddings - backed by Attorney-General George Brandis and Resources Minister Matt Canavan - was rejected 25 to 38 just after 11pm.

Labor had earlier in the day announced it would vote as a bloc against all amendments, including those from the Greens, One Nation and David Leyonhjelm to be debated on Wednesday.

In a moving speech, Senator Brandis marked the bill's passage as "both an expiation for past wrongs and a final act of acceptance and embrace" of gay and lesbian Australians, particularly the young.

"By passing this bill, we are saying to those vulnerable young people: there is nothing wrong with you. You are not unusual. You are not abnormal. You are just you," Senator Brandis said.

"There is nothing to be embarrassed about. There is nothing to be ashamed of. There is nothing to hide. You are a normal person and, like every other normal person, you have a need to love."

Marriage equality would be seen by history as one of the Turnbull government's signature achievements, Senator Brandis predicted. "It rises above tawdry day-to-day politics as an imperishable legacy," he said.

Tuesday's debate marked the 23rd time a same-sex marriage bill has been brought to the Parliament, but only the fourth time one reached a vote. The previous bill in 2012 was defeated in the Senate 26 votes to 41.

Despite earlier suggestions of a protracted debate about religious freedom, the bill progressed quickly through the Senate on Tuesday evening, and it seemed likely to pass essentially unchanged on Wednesday.

Labor locked its MPs into rejecting all the amendments, including those from Liberals and the Greens, arguing the bill by Liberal senator Dean Smith already represented a cross-party consensus.

Even the half-dozen socially conservative Labor senators, mostly linked to the Shoppies union, agreed to oppose the amendments on the basis that religious freedoms could be dealt with separately next year.

Nonetheless, several Labor senators including Helen Polley from Tasmania, Don Farrell from South Australia and Chris Ketter from Queensland indicated they would exercise their right to vote against the final same-sex marriage bill.

Early on Tuesday afternoon, amendments moved by Liberal senators James Paterson and David Fawcett were defeated, including one that would have inserted two definitions of marriage into the law: one being "a man and a woman" and the other being "two people".

Moves to allow civil celebrants to discriminate against same-sex couples were also defeated. In doing so, Labor's Senate leader Penny Wong said it would be a "very unorthodox" and "illiberal" step to exempt people from anti-discrimination laws based on conscientious beliefs alone.

Coalition senators had a conscience vote, and several frontbenchers joined Labor and the Greens to vote down the conservatives' amendments, including Defence Minister Marise Payne, Education Minister Simon Birmingham and Indigenous Affairs Minister Nigel Scullion.

Other proposals - including a law to allow parents to remove their children from schools that don't teach the traditional definition of marriage, and an "anti-detriment clause" to ensure doctors and teachers are not deregistered for their beliefs - were also defeated on Tuesday night.

The expedited passage through the Senate means a final vote as soon as Wednesday is likely. But because of the Turnbull government's decision to cancel this week's session of the House of Representatives, the bill cannot proceed to that chamber until next week.