Giffords first spoke within the past few days and is speaking "more and more," spokesman C.J. Karamargin said Wednesday. He didn't know what her first words were, but said that at breakfast one morning she asked for toast.
"She's working very hard and it's paying off," he told The Associated Press. "We're elated at this. We always knew Gabby is a fighter and that she's not going to let this thing win. And you know, every day is proof of that."
Six people, including a 9-year-old girl and a federal judge, were killed in the attack outside a grocery store where Giffords was meeting with constituents. Thirteen people, including Giffords, were injured.
Other news organizations, including Politico, earlier reported that Giffords had asked for toast and was able to speak.
In a Facebook post on Tuesday, Giffords' husband Mark Kelly said his wife had her appetite back and was eating three times a day, "even though it's hospital food."
"It is hard to believe that only one month has passed since Gabrielle was shot," he wrote. "The doctors say she is recovering at lightning speed considering her injury but they aren't kidding when they say this is a marathon process."
He said "there are encouraging signs every day," pointing to her renewed appetite.
"Your prayers are being heard, so don't stop," he wrote.
Kelly, a NASA astronaut, said last week that he expects his wife to be well enough to be at his space launch in two months.
The space shuttle Endeavour will leave April 19 for a two-week mission to the International Space Station, and Kelly will be on board leading a veteran, all-male crew. The mission will be Endeavour's final flight and Kelly's fourth.
Dr. Gerard Francisco, who is treating Giffords at a rehabilitation facility in Houston, said Tuesday that he hopes the congresswoman can make enough progress to attend the space launch, but said it's too early to say.
Arizona state Sen. Ken Cheuvront, a friend of Giffords who shared a house with her and another lawmaker when they served together in the Legislature, said he was excited to hear that she is speaking again.
"It gives all of us a lot of hope that she'll have a full recovery," Cheuvront said. "She's a fighter, there's no ifs, ands or buts about that. She's always been very tenacious, and I have no doubt whether it's politics or her recovery that she'll still have that same amount of energy."