And the government's leader in Senate, Attorney-General George Brandis, saying he wanted to see further protections for people who did not want to be involved in same-sex marriages.
It also seeks to protect freedom of speech and enacts a narrow anti-detriment clause, which would prevent governments and agencies taking adverse action against someone with a traditional view of marriage.
"There's an inevitability about the passage of this bill", Senator O'Sullivan said.
Mr Brandis said parliament would start the debate on same-sex marriage legalisation with Senator Smith's bill, if a "yes" vote is victorious.
It also includes exemptions so religious organisations can refuse to conduct same-sex marriages.
The starting point for debate will now be the cross-party bill authored by the Liberal senator Dean Smith and supported by Labor, the Greens and others.
"But I suspect it will need improvement in terms of strengthening religious protections", he told ABC radio.
Once the bill clears parliament, it requires the governor-general to give it royal assent.
"It's already against the law", Senator Brandis said.
Australia voted in favour of same-sex marriage today, with 61.6% of the almost 13 million people who voted in the postal survey saying yes.
The national director of Liberals and Nationals for Yes, Andrew Bragg, has said a marriage bill should be guided by three principles: "Firstly, existing discrimination in the Marriage Act should be eliminated; secondly, a strong protection for religious freedom should be provided; and thirdly, we should not reintroduce commercial discrimination in Australia".
"Extending marriage to same-sex couples will strengthen and not diminish marriage in Australia".
The Human Rights Law Centre said Paterson's bill was "a Trojan horse to allow unprecedented discrimination".
Liberal National Party senator Barry O'Sullivan said the nearly five million Australians who voted "no" were cautious about changing the definition of marriage.
The ABS will announce the results of the survey at 10am AEDT tomorrow.
Some 22 bills seeking same-sex marriage have been introduced into the federal parliament since 2004.
In contrast, Smith's bill allows exemptions only for religious organizations and ministers. We know that's the real slippery slope, when you unravel anti-discrimination protections, and I don't think Australian people want that.
'That's the goal, ' Prime Minister Malcolm Turnbull vowed ahead of the Australian Bureau of Statistics announcing on Tuesday which way more than 12 million people voted on the issue.
"If you are a gay man or a gay woman and you go into a florist and say "I'd like to buy a bunch of flowers", it's just wrong and illegal for florist to say "I don't" serve gay people" just as it would be wrong or illegal for the florist to say to an indigenous person "I don't serve indigenous people'".