The acquisition will make Sky the second largest provider in the UK broadband market.
02's home phone and broadband service currently has over half a million customers and will become Sky customers after the acquistion.
"Sky has been the UK's fastest-growing broadband and telephony provider since we entered the market six years ago. From a standing start in 2006, we have added more than 4.2 million broadband customers," said Jeremy Darroch, Sky's Chief Executive.
"The acquisition of Telefónica UK's consumer broadband and fixed-line telephony business will help us accelerate this growth. "We believe that the O2 and BE consumer broadband and telephony business is a great fit, with customers used to high-quality products and strong levels of customer service. We look forward to welcoming these new customers to Sky and giving them access to our wide range of high-quality products, great value and industry-leading customer service."
Sky will pay £180m to Telefónica UK for O2 and BE brands. After the acquisition is complete O2 and BE customers be migrated into Sky's unbundled network.
Analysts say that the acquisition will now give BSkyB an edge over its main competitor.
"The news of the takeover is significant on several fronts. Firstly, it makes BSkyB the second largest broadband provider in the UK," said Emeka Obiodu, telco strategy analyst at Ovum. "This strengthens the hand of BSkyB in the market as the company would now be able to boast of having a bigger TV customer base, and more broadband customers than its main rival - Virgin Media. Such a prospect will concern Liberty Global a bit when it moved to take over Virgin Media in February."
Obiodu says the sale however, reduces the power of UK mobile operators in the converged space.
"At a time when telcos across Europe are intensifying efforts to offer converged (fixed, mobile, broadband, TV) services, mobile telcos in the UK are making themselves much more reliant on mobile," he added. "EE still retains a presence in the fixed telecoms space but its offering is increasingly infrastructure-light. Ultimately, this is a dangerous scenario as it might reduce the strategic maneuvererability of the UK's mobile telcos in a converged future."
The acquisition is due to be finalised by the end of April 2013.
BSkyB currently has 4.2 million broadband customers and 4 million telephony customers.