In 1979, former Liberal party leader Thorpe was tried and acquitted of conspiring to murder his ex-lover Norman Scott.
Newspapers at the time described it as "the trial of the century".
A Very English Scandal is based on the bestselling book by author and former journalist John Preston.
Thorpe was the first British politician to stand trial for conspiracy and incitement to murder.
In the mid-70s, stories circulated about Thorpe's relationship with a former male model, Norman Scott.
The relationship was alleged to have started in 1961, when male homosexual acts were illegal.
The story broke when Scott was appearing at a court in Barnstaple on a minor social security charge.
During the hearing, Scott shouted out, "I am being hounded because of my sexual relationship with Jeremy Thorpe."
Thorpe issued an immediate denial but when an affectionate letter between them appeared in the press, Thorpe resigned as leader.
But worse was to come when Thorpe was accused of plotting to kill Scott.
Thorpe was arrested, charged and later acquitted, but his career was in tatters.
Grant, star of films such as Four Weddings and a Funeral and Notting Hill, recently collaborated with Frears in the critically-acclaimed Florence Foster Jenkins for which he earned Golden Globe, Bafta and Screen Actors Guild nominations.
Piers Wenger, controller of BBC Drama, said: "Hugh Grant is an extraordinary actor and a worldwide star. I'm delighted that he is joining the BBC One family and I can't wait to see how he applies his unique talent bringing to life this mercurial role."
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Hugh Grant is to play disgraced MP Jeremy Thorpe in a three-part BBC TV drama written by Russell T Davies and directed by Stephen Frears.