The production, which features Sherlock star Andrew Scott in the title role alongside Jessica Brown-Findlay and Juliet Stevenson, ends its run at the Harold Pinter Theatre on 2 September and will be beamed into living rooms around the country in 2018.
Icke said: "I am very much looking forward to this next step on BBC Two. To be able to offer our version of Hamlet to as wide and diverse an audience as possible has always been of paramount importance to us, and now we are thrilled to be able to bring it to people across the country."
In her five-star review for WhatsOnStage, Sarah Crompton said Scott provided "an outstanding performance on a truly memorable night."
Hamlet runs at the Harold Pinter until 2 September.