The play has been written by John Logan and will be directed by Michael Grandage when it opens in London in March 2013.
The story is based on a true encounter between the pair but is a fictionalised account of their one brief meeting.
Michael explained to the Daily Mail newspaper: “You have the real life figures of Alice and Peter who met each other briefly for about 40 minutes. John Logan imagined what they said to each other and that’s the basis of the play.”
He admitted it will be a dark play which poses questions about how people cope with being hurled into the public eye as children, asking “Would you prefer it never happened?” and added “It’s a mature, quite dark drama.”