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On-again, off-again couple Nicole Richie and Adam "DJ AM" Goldstein are off again.

"We are confirming that we have amicably separated," the pair said in a statement released by Richie's publicist, Cindy Guagenti.

Richie, 24, and Goldstein, 33, began dating in 2004 and announced their engagement a year later in February 2005. But in early December, Richie's rep confirmed that the wedding was off.

But by March of this year, the pair were an item again, holding hands all over Los Angeles and even vacationing together at the Las Ventanas resort in Los Cabos, Mexico.

"They decided to try again," a friend told PEOPLE at the time. "They never stopped caring about each other." Although Richie hadn't put her engagement ring back on, the friend said, "they really are so happy together. ... It was only a matter of time before they got back together."

Richie even threw Goldstein a surprise 33rd birthday party at Pure Nightclub in Las Vegas recently, where she toasted him by saying, "This is (Adam's) eighth year of sobriety. I'm so proud of you. You've come so far and I love you so much."

On Wednesday, Goldstein was in New York City to DJ at the Entertainment Weekly Upfronts Party in New York. Said an observer: "He was keeping to himself at the party … but he was working. He didn't look particularly happy, but he could have just been concentrating."

Richie, meanwhile, is said to be at home in L.A. gearing up for a press tour promoting the new season of The Simple Life and working on her first album and a follow-up to her novel The Truth About Diamonds.

Richie recently blamed her dramatic weight loss on, in part, her December breakup with Goldstein. "I get really stressed out, and I do lose my appetite," she told Vanity Fair, adding that she's since sought professional help.