The company, which displays a 'By Royal Appointment' logo on its homepage, will pay £250m upfront and the remaining £120m in three instalments over 18 months.
The deal is subject to EU competition authority clearance, and completion is expected in December 2006.
"I think that Carphone Warehouse was a bit naïve in underestimating the operational difficulty of running an ISP and transitioning customers to the unbundled network," Fogg told vnunet.com.
"It is interested in building a telephony-based business, and broadband is a way of doing that."
Carphone Warehouse boss Charles Dunstone admitted earlier this month that the company's free broadband offer had generated more consumer demand than the company could satisfy.
The company has received 625,000 applications for the service, but provisioned for just 421,000.
Carphone Warehouse was said to be up against Sky in the bidding for AOL, but Fogg suggested that there would have been more "tension" in the Sky negotiations.
"Sky was smart to buy EasyNet last year, then waited until it offered the free broadband service," he said.
"Sky is an entertainment company, and the broadband business is a way of delivering content. It would not have been such a natural fit with AOL, which had its own content."
A joint AOL and Time Warner statement said: "This transaction marks an important step in positioning AOL's Audience business to take further advantage of the rapid growth in online advertising in keeping with AOL's overall worldwide strategy."
Dunstone added: "The joint development of AOL's already successful Audience platform will bring us new advertising and content revenues in a proven and low risk manner."
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