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Sponsors of the bill tried to move through similar legislation in 2011, but met with opposition and were forced to change the bill to allow civil unions only.
Rhode Island remains the last of New England's six states without a law allowing gay and lesbian couples to wed, after Maine legalized same-sex marriage late last year.
Another 31 states have passed constitutional amendments restricting marriage to heterosexual couples.
In Rhode Island, it remained unclear how the proposal would fare in the state Senate, though Handy said that in the last election cycle, both legislative chambers had elected more "pro-equality" candidates than ever before.
The state Senate president, Teresa Paiva Weed, opposes gay marriage but has said she would allow a Senate Judiciary Committee vote on the bill if it passed in the House.
Senate spokesman Greg Pare said earlier on Thursday that predictions of a close Senate vote had not changed.
(Reporting by Daniel Lovering; Editing by Jane Sutton and David Brunnstrom)