Fox's bid to buy the 61% of Sky it does not already own had been held up by regulators amid concerns the US company would have too much influence over the UK media landscape.
Final approval of the deal by newly appointed Culture Secretary Jeremy Wright ends a regulatory saga over the tie-up which was first announced in December 2016.
To address those concerns, Fox has agreed to sell Sky News to Disney, with a commitment that the channel would be funded to the tune of £100m a year for the next 15 years.
Mr Wright said this marks "the final stage of the public interest consideration of this case" and that it is "now a matter for the Sky shareholders" to decide whether to accept Fox's bid.
Last month, the government indicated that it was prepared to allow the takeover under these conditions.
Fox's biggest shareholder is the Murdoch Family Trust, which is also the biggest shareholder in News Corporation, owner of The Sun, The Times and The Sunday Times.
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Sky is at the centre of a transatlantic takeover tussle between Mr Murdoch's Fox and Comcast, which owns NBC, Universal film studios and E!.