Debate in the ACT Legislative Assembly returned to gay rights on Tuesday, as members passed a motion marking the first anniversary of the territory's historic same-sex marriage law passed on October 22, 2013.

Moved by Labor backbencher Yvette Berry, the motion called on the federal Parliament to end discrimination against same-sex couples after the High Court overturned the law in December.

The debate comes as the Lloyds prepare to travel to a conference in the US city of Baltimore next week, where they will also register and complete a civil same-sex marriage.

They form one of just 31 same-sex couples who married before the territory law was found to be inconsistent with the Commonwealth Marriage Act and was thrown out.

After more than a decade together, the public servants have also celebrated registering their civil partnership.

"Our first wedding had four TV news crews and six photographers," Ms Lloyd said. "We feel as married between ourselves as we did before but it's still disappointing not to be recognised the same as everyone else."

"I think it is really important that people understand that the legal difference still exists."

Opposition Leader Jeremy Hanson and Liberal members voted against the motion on Wednesday, remaining at odds with recent polling that showed as many as 72 per cent of Australians support same-sex marriage and want Prime Minister Tony Abbott to allow a conscience vote for federal Liberal MPs.

Ms Berry said 17 countries around the world allow same-sex couples to marry, including New Zealand, Mexico, the United Kingdom and 30 American states.

Treasurer Andrew Barr slammed Mr Hanson and opponents of same-sex marriage, saying they allowed discrimination to exist in Australian society and would be found on the wrong side of history.

"It absolutely overdue," Ms Lloyd said. "It's almost a non issue for most Australians and people are surprised when they find out we still can't get married here."

Couples married in the ACT last year used a private online forum called 'Canberra's First 31' to connect and monitor developments in politics of same-sex marriage in Australia and overseas.

Ms Lloyd said the group formed critical support group for the families.

"We went through lots of phases of grief and anger and disbelief, discussing what we were going to call our partners, our husbands and wives."

Jennifer Lloyd used a post in the group to express frustration with the lack of reform, saying she remained as eager as ever to be married to her wife.

The ACT ministerial advisory council welcomed moves to mark the anniversary and renewed push for equality.

"This landmark law reform, the first in Australia, affected not just the 31 couples who married in the ACT, but also their families, friends and community," council member Anne-Marie Delahunt said..

"The nation was watching, and it saw that the ACT was ready and anxious to see same-sex couples treated with the same dignity and status as other couples who can marry."