Under the agreement, Worldwide - the BBC's commercial arm - will pay Lonely Planet founders Tony and Maureen Wheeler AUD$67.2m (£42.1m) to complete the full takeover.
The deal follows a put option over the remaining shares, jointly agreed between Worldwide and the Wheelers at the time of Lonely Planet's majority acquisition in 2007.
"The put option enabled us to benefit from the Wheelers' experience over the last three-and ahalf years," said Marcus Arthur, chairman of Lonely Planet and managing director of BBC Worldwide global brands.
"They have supported Lonely Planet's ongoing migration from a traditional book publisher to a multi-platform brand. I would like to wish them the very best in the next phase of their lives."
In a joint statement, Tony and Maureen Wheeler said: "The last three years have seen Lonely Planet embark on a journey of its own - giving its users and readers ever more choice and utility. We wish the business and the Lonely Planet community every success in the future."
Worldwide paid £88.1m in October 2007 for a 75% stake in Lonely Planet, but the move was instantly criticised within the industry on the grounds that the BBC was going outside of its core remit.
In his MacTaggart lecture at the 2009 Edinburgh Television Festival, News Corporation's James Murphy even described the acquisition as a "nationalisation" of the travel guide publisher.
Worldwide's management have consistently defended the deal, but the BBC Trust ruled in 2009 that Worldwide must not undertake similar acquisitions again in future, unless there are "exceptional circumstances".