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Rea Carey, executive director of the National Gay and Lesbian Task Force, called on the Senate to pass the measure quickly.

“The cruel irony of the Supreme Court’s recent marriage decisions is that an LGBT couple could get married one day, and on the very next day, because we still don’t have federal laws to ban employment discrimination, those same individuals could be fired from their jobs,” Ms. Carey said.

About 88 percent of Fortune 500 companies have already adopted non-discrimination policies that include sexual orientation, according to the Human Rights Campaign. About 57 percent of those companies include gender identity.

Mr. Alexander said the GOP planned to offer several amendments to the bill before it comes to a floor vote, including one that would require the Equal Employment Opportunity Commission to define when a person is “in transition” to change their gender identity.