Contradicting rumors that it would sell Skype back to its founders, eBay announced today that it would separate the Internet calling business from eBay’s core auction business through an initial public offering in the first half of 2010.

A lot can happen between now and then, including bids from private equity firms to buy the division, which eBay bought for $2.6 billion 2006.

But John Donahoe, chief executive of eBay, said that Skype is a great stand-alone business with strong fundamentals and accelerating momentum. He acknowledged that the expected synergies between Skype and eBay didn’t happen. The IPO will give Skype a chance to focus more resources on expanding its presence in online voice and video communications.

Last week, rumors surfaced that Skype founders Niklas Zennstrom and Janus Friis were trying to buy back the company. Donahoe, who joined eBay in April, 2008, had said that the company would spend a year evaluating Skype. He installed a new leader, Josh Silverman. Skype revenues in 2008 were $551 million, up 44 percent. There are 405 million users, up 47 percent from 2007. Skype revenues are expected to surpass $1 billion in 2011.

The company said that the recent release of Skype for the iPhone has generated a great response. In the first 36 hours after its release, it became the No.1 downloaded free iPhone app in more than 40 markets. In a week, downloads have topped two million.

It’s been a week of departures and refocusing at eBay. The Skype news comes barely a day after social news discovery site StumbleUpon, another company that eBay had acquired, announced that it was spinning off to become a private company again.

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