Australia’s first same-sex marriage has taken place a week after the historic bill was passed.

Lauren Price, 31, and Amy Laker, 29, made history when they tied the knot in Sydney on Saturday.

The pair were married at an outdoor ceremony in front of 65 guests after being engaged for two years.

They had initially planned a civil union but were granted an exemption from the mandatory 30-day waiting period.

The first same-sex weddings were expected to begin on January 9 next year, after Australia’s parliament passed a bill legislating for marriage equality on December 9.

This can be waived in exceptional circumstances, as it was for Amy and Lauren as they had relatives travelling from Wales.

Another exemption was granted to Amy, 36, and Elise McDonald, 28, who also had guests travelling from Asia and the United Arab emirates.

The McDonalds, who were engaged for two years after meeting six years ago, exchanged vows in Melbourne's Carlton Gardens.

Elise told the Herald Sun: “We’re just so excited that it’s been legalised and we have the support of the country and everyone is behind us.”

Fellow Melbourne couple Heather Richards, 56, and Cas Willow, 53, have also been granted an exemption to marry on Monday.

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The pair were allowed to wed as Ms Willow is suffering from breast cancer, which has spread to her brain.

“I don't even know if I'll make it to Christmas, let alone January 9, so they allowed us to get married early”, she told ABC news.

Staff at the Peter MacCallum Cancer Centre, where Ms Willow is a patient, are planning for the ceremony.

The hospital’s chief executive, Dale Fisher, said they were “thrilled” to support their big day.

The women decided against having a commitment ceremony, opting for marriage, and got engaged when the law was changed.

Ms Richards told the website: "It means our relationship won't just be tolerated, it will be accepted.”

Australia has become the 25th country to recognise same-sex marriage.

When the vote was passed, Prime Minister Malcolm Turnbull said: “We’ve voted today for equality, for love, it’s time for more marriages, more commitment, more love, more respect.