The drama is based on the best-selling book of the same name, which was written by former Arts Editor of the Evening Standard and Sunday Telegraph, John Preston.
Davies is writing the adaptation, which is being directed by Stephen Frears.
In a statement T Davies said: "It’s a dream come true, to work with Ben. One of Britain’s finest actors playing one of history’s most fascinating men.
Piers Wenger, “Ben Whishaw is an extraordinary actor and we are thrilled to announce he will star opposite Hugh Grant in this scandalous real-life story for BBC One.
"I cannot wait to see this first-class British duo bring Russell’s powerful scripts to life."
Whishaw most recently appeared in an episode of Queers (The Man on the Platform) , where he performed one of a series of monologues about the British gay experience.
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