SPRINGFIELD, Ill. (CBS) — The House sponsor of legislation to legalize same-sex marriage in Illinois said the measure could get a vote “very soon,” after the new General Assembly was sworn in this week.

State Rep. Greg Harris (D-Chicago) said, when same-sex civil unions became law in 2011, it was described as the first step in a very long journey toward legalizing gay marriage. But Harris said that has changed.

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“We are on a fast track to passing marriage equality in the state of Illinois,” he said.

Harris said a new measure on same-sex marriage is coming “very soon.” He said sentiment in support of same-sex marriage has been growing quickly in recent years.

“The people of America are ahead of us,” said Harris. “They are doing the right thing. They believe the right thing. The journey is short, because people want it to be short. The journey to full marriage is not going to be as long as we thought, because folks have talked to their neighbors, they’ve talked to their clergy people; they’ve decided treating every couple equally in the eyes of the law is the right thing to do.”

The drive to legalize same-sex marriage in Illinois stalled last week when it became clear there would not be enough votes for passage during the lame-duck session of the previous General Assembly.

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Harris said more new, younger lawmakers in Springfield could be open to voting for the measure, and could have the kind of perspective to make a same-sex marriage vote a majority.

“If you look in the House, you’ve got a lot of people who are younger, and folks who are younger see this as a vote of true equality. It’s something they’ve grown up with, it’s something their kids talk to them about when they come home from school, it’s what people believe in,” Harris said.

That doesn’t mean support for same-sex marriage would be a landslide. The head count was so close in the Senate last week that it took only two senators absent for family illnesses and a third senator on vacation to make a difference. The bill did not move to a floor vote in the Senate.

Under the new General Assembly, Democrats have increased their majorities in both chambers, which could make passage more likely.

Illinois Senate Minority Leader Christine Radogno (R-Lemont) also signaled last week that there could be bipartisan support for same-sex marriage if legislation is tweaked to address concerns about whether churches opposed to gay marriage would be forced to allow their facilities to be used for such ceremonies.

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Harris also noted hundreds of ministers and other religious leaders from across the state have voiced support for same-sex marriage in recent weeks.