Colorado-based Liberty confirmed in a statement that American Chase Carey had been appointed chairman and chief executive of F1, with Ecclestone moving into an advisory role as "chairman emeritus".
Ecclestone, 86, had already signalled the end of his near 40-year reign as head of Formula One in comments to German magazine Auto Motor und Sport earlier Monday.
"I was dismissed today," Ecclestone told the magazine. "I no longer run the company. My position has been taken by Chase Carey," he said.
Ecclestone, a former car salesman who transformed F1 into a multi-billion dollar business, was initially reported to be staying on after news of the Liberty deal emerged last year.
However in a statement confirming completion of the takeover, valued at around $8 billion, Liberty president Greg Maffei confirmed Ecclestone would be leaving his role as chief executive.
"We are delighted to have completed the acquisition of F1 and that Chase will lead this business as CEO," Maffei said.
Carey, a vice-chairman of the 21st Century Fox media conglomerate, said he was looking forward to running a sport which had "huge potential with multiple untapped opportunities."
"I have enjoyed hearing from the fans, teams, FIA, promoters and sponsors on their ideas and hopes for the sport," Carey said.
"We will work with all of these partners to enhance the racing experience and add new dimensions to the sport and we look forward to sharing these plans overtime."
"The sport is what it is today because of him," Carey said of Ecclestone. "I am grateful for his continued insight and guidance as we build F1 for long-term success."
McLaren executive director Zak Brown was among the first to lead the tributes to Ecclestone.
"Today we should all pay tribute to a remarkable entrepreneur," Brown said on McLaren's Twitter feed. "He will be a very hard act to follow."
In separate comments on the pitpass.Com website, Brown said Ecclestone had turned Formula One into a "sporting powerhouse."
The takeover gives Liberty, which is backed by US media titan John Malone, control of a global and highly profitable sport that rakes in billions from advertisers and broadcasting rights.
Formula One-branded merchandise also brings in millions, but some F1 teams are plagued by financial problems and the sport has limited activity in the social and digital media platforms crucial to courting the next generation of fans.
Carey has a proven record in expansive sport-media growth, and expertise in the value and exploitation of sports rights, notably in the US market.
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