Prime Minister Scott Morrison has "committed to getting an outcome" on constitutional recognition of Aboriginal and Torres Strait Islander peoples, paving the way for a national discussion on the best way to achieve it.
But he has given no timeframe on how long the process might take.
The freshly elected Mr Morrison told the Herald that "we need to work together across the aisle and across our communities to get an outcome that all Australians can get behind and we'll take as long as is needed to achieve that."
Labor had committed to a referendum in the current parliamentary term had it been elected.
Mr Morrison said "my priorities for Indigenous Australians are to ensure Indigenous kids are in school and getting an education, that young Indigenous Australians are not taking their own lives and that there are real jobs for Indigenous Australians so they can plan for their future with confidence like any other Australian."
His comments coincide with his appointment of Western Australian MP Ken Wyatt as the country's first Indigenous cabinet minister, with the title of Minister for Indigenous Australians.
Senior Aboriginal and Torres Strait Islander leaders gathering in far north Queensland this weekend called for a meeting with Mr Morrison "as soon as possible" to try to make progress on constitutional recognition.
They were marking the second anniversary of the solemnly-worded Uluru Statement from the Heart, which was adopted in May 2017 at an unprecedented summit of Indigenous leaders from around the country. It issued a plea for a "First Nations Voice [to be] enshrined in the constitution" to "empower our people and take a rightful place in our own country."
The communique of around 40 Indigenous leaders at the weekend welcomed the Liberal party's promise made during the election campaign of $7.3 million to develop a proposal to take to a referendum, and the budget allocation of $160 million to bring that referendum about.
But it said the way forward "must be informed by Aboriginal and Torres Strait Island people throughout Australia", saying the movement for recognition was "growing, and will continue to grow".
Mr Morrison’s predecessor Malcolm Turnbull dismissed the idea of the Voice saying it would amount to a third chamber of parliament.
A Liberal party policy document released in the last days of the campaign said more work was needed on "what model we take to a referendum and what a Voice to parliament would be.” It talked broadly of “comprehensive co-design of models to improve local and regional decision making and options for constitutional recognition."
Indigenous leaders are rapidly re-calibrating expectations after the shock election victory of the coalition.
On Sunday morning, many travelled to the historic Tree of Knowledge at Yarrabah, outside Cairns, a site closely associated with the launching of the successful 1967 referendum which gave Indigenous Australians the right to be counted in the census and allowed federal laws to be made for Aboriginal and Torres Strait Islanders.
Monday marks the 52nd anniversary of that referendum – one of only 8 to be carried out of 44 referendum proposals ever put to the country.
The leaders' weekend communique said the country united in 1967 and "we can do it again … We invite all Australians to walk with us on this journey, thoughtfully and with purpose, and to support our voice being heard.”
A bipartisan parliamentary committee last December recommended that the Voice “should become a reality”, after a co-design process between government and First Nations peoples.
A range of industry and other organisations have also come out in support of the Uluru statement, including the Law Council of Australia, the AMA, the business council, ACOSS, major law firms, big miners BHP and Rio Tinto and – as of last week – 21 leaders of investment banks, super funds and accounting firms.
Some Indigenous leaders told the Herald on Sunday that Mr Morrison’s re-election might not slow the momentum for constitutional recognition.
Pat Turner, co-chair of a new joint council formed between Indigenous peak bodies and state and federal governments to re-design 'Closing the Gap' targets, said she had been impressed with the Prime Minister so far.
"This is a historic agreement that he reached with us [on 'closing the gap' between indigenous and non-indigenous well-being] and because of his superb leadership on it we are going to work closely with him" Ms Turner said. She added that constitutional recognition was a "complementary parallel process" and "it's important both get done."
Lawyer and human rights advocate Teela Reid, a Wiradjuri and Wailwan woman, told the Herald that the election result "does not really change much for blackfellas. We have always had this kind of experience with the left and the right. History has proven that change is not easy."
She said if anything, a re-election of a coalition government "motivates us more". But she said the Morrison government should get a referendum done "as soon as possible. Our old people don't have time, they deserve the question to be resolved in their lifetimes".
Many of those behind the Uluru statement want a referendum to embed the principle of a Voice first, with detailed design taking place after a successful vote. But some indigenous leaders, such as Tom Calma, co-chair of Reconciliation Australia, worry that a too-vague proposal will not command majority support.
A government source told the Herald, “The Prime Minister is intensely pragmatic. He will get a result on this. He just wants the right one.” A referendum without an agreed model risked getting "very Brexity" the source said, adding that a fixed timeframe could put pressure on a fragile process.
Labor's putative leader-elect Anthony Albanese told the Herald that "if there is one area where we can put aside partisanship and work together in the national interest, it must be to advance the agenda of the Uluru statement."
Mr Morrison said recognition must be achieved alongside "practical goals" which made Indigenous Australians "safe in their communities" and enjoying the same access to services as any other Australian.