Elliott, who was the morning show's news anchor, will be replaced by Amy Robach, ABC announced in a memo obtained by PEOPLE.
ABC News president Ben Sherwood spoke of the move in the memo to staff on Sunday night, confirming that he'd been in contract talks with Elliott for several months and while "we worked hard to close a significant gap between our generous offer and his expectations … Josh felt he deserved a different deal and so he chose a new path."
Added Sherwood: "I want to thank Josh for his many contributions to GMA and ABC News."
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"We always knew that Amy was special and we have all been especially inspired to watch her battle breast cancer with grace and determination," Sherwood said.
"Indeed, she thrilled us with two weeks of memorable dispatches from Sochi anchoring our Olympic coverage – all between chemo treatments. Amy will be a fierce and formidable full-time addition to our GMA team."
Elliott's is the second departure from the show in recent months. Meterologist Sam Champion left the show in December.