The production will play at London's Bridge Theatre for eight weeks from 10 September, organisers announced on Wednesday.
"I grew up being at the theatre a lot, my mother [Maria Fahl] was a stage actress, and I think even when I was dreaming of becoming an actress myself, being on stage was the journey that I kind of visualised," Vikander said.
"Back in Sweden, where I'm from, if you're an actor then really what you are is on stage. And you're lucky to maybe have a TV show or film every couple of years, because that's how small the industry is in Sweden.
"So I think that's what I always saw in front of me. And then life happened, and throughout the years [theatre] has always been something I've been waiting for and thinking 'it will happen'."
The play has been adapted and directed by Simon Stone, and will also star The Walking Dead's Andrew Lincoln.