The government has finally secured the repeal of medevac by coming to an arrangement with crossbencher Jacqui Lambie, the terms of which she refused to disclose to the Senate because of “national security concerns”.

The repeal was carried 37-35 after Lambie – on whose vote the result depended – said she was satisfied with her negotiations with the government.

But Senate leader Mathias Cormann had previously unequivocally denied any deal.

Prime Minister Scott Morrison personally negotiated with Lambie over the repeal, a sign of the government’s determination to quash the medevac law.

Medevac – which facilitated medical transfers of people from Papua New Guinea and Nauru - was passed against the Coalition’s opposition when it was briefly in minority late in the last parliamentary term.

Lambie had indicated previously that her vote for the repeal would be conditional on the government meeting a condition which she would not specify. She suggested security matters were involved.

In an emotional speech Lambie, breaking down in tears, told the Senate: “I’m not being coy or silly when I say I genuinely can’t say what I proposed. I know that’s frustrating to people. And I get that. I don’t like holding things back like this.

"But when I say I can’t discuss it publicly due to national security concerns, I am being 100% honest to you. My hand is on my heart and I can stand here and say that I would be putting at risk Australia’s national security and national interest if I said anything else about this,” she said.

“I put a proposal to the government, and since then we have worked together really hard to advance that proposal. We’ve worked to an outcome I believe we both want, which is an outcome where our borders are secure, the boats have stopped and sick people aren’t dying while waiting for treatment.

"And as a result of that work, I am satisfied, I am more than satisfied, that the conditions are now in place to allow medevac to be repealed.”

Read more: Peter Dutton is whipping up fear on the medevac law, but it defies logic and compassion

But Cormann earlier said: “There is no secret deal. There will be no change to our strong border protection arrangements. There will be no change to our strong national security arrangements. And there will be no change in the way we deal with the legacy caseload that Labor left behind.”

Greens leader Richard Di Natale demanded to know who was lying. “Who’s lying, minister Cormann? Are you lying? Or is senator Lambie lying?”

He said Cormann had “walked over to senator Lambie and said, ‘Is it OK if I say there’s no deal?’ We heard you say it”.

One Nation’s Pauline Hanson told the Senate she had just had “a quick talk with senator Lambie”. It was “extremely hard for her. … I do trust her judgement”.

When the Senate resumed on Wednesday morning, the government immediately moved to bring on the repeal bill. Labor demanded the terms of the deal, declaring the vote should be delayed until they were revealed.

Labor’s leader in the Senate Penny Wong said: “We have cabinet ministers coming in here like lemmings, voting on a legislation based on a deal you haven’t seen.”

Labor spokeswoman on home affairs Kristina Keneally said the Australian public supported medevac, quoting a poll showing 62% in favour. She said medevac had nothing to do with Operation Sovereign Borders which Labor supported. Repeal would deny people treatment, she said.

Under the offshore arrangements, there are currently just over 200 people in Papua New Guinea and more than 250 on Nauru.

At a joint news conference with Home Affairs Minister Peter Dutton, Morrison said: “The only undertaking we’ve given is to implement our policies, that is it”.

Amid speculation about some resettlement undertakings, Morrison said the government’s policies sought to resettle people.

Michelle Grattan does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.