The actress will star as fading belle Blanche DuBois in Sarah Frankcom's production at the Manchester Royal Exchange.
Peake will join forces with Frankcom once again, following her portrayal of Hamlet in 2013 and after she starred in The Skriker. She has also worked with the director on The Masque of Anarchy.
Williams' play won the Pulitzer Prize for Drama in 1948 and follows Stella and husband Stanley in New Orleans as they are visited by Stella's difficult sister Blanche DuBois. DuBois arrives from Mississippi after her family home is lost to creditors.
Peake said: "Blanche DuBois is a monumental woman, complex, multi-layered and beautifully painted by Williams in this staggering play. Finding out who she is and developing her in a rehearsal room with a brilliant and generous director like Sarah is a thrilling journey to go on. I can’t wait to start."
A Streetcar Named Desire runs at the Manchester Royal Exchange from 13 September, with previews from 8 September.