Warchus, who is currently an artistic associate at the theatre, will take up the role in the autumn of 2015 when Spacey, currently in his tenth year as artistic director, steps down.
"I couldn't be happier that my friend Matthew Warchus has agreed to take the reins at the Old Vic next year," said Spacey.
"I have long admired his work and had a wonderful experience when we worked together on Speed-the-Plow. Matthew is a thoughtful, intuitive and highly creative director and he has rightly been applauded for his work, in particular the quality and diversity of his portfolio."
Warchus responded: "I am excited and honoured to be following Kevin's galvanising tenure at this wonderful building. He has re-established the Old Vic as a globally important theatre and I look forward to continuing to develop it as a hub of invigorating creativity."
Kenneth Branagh had previously been heavily tipped to succeed Spacey, and today's announcement came out of the blue - no press conference was held, and Warchus' name wasn't known to be in the frame.
Nick Clarry, chairman of The Old Vic Theatre Trust, commented: "This is a key appointment for the Old Vic, building on the many achievements of Kevin Spacey since 2004. We believe that the next few years will be a very exciting time."
Clarry revealed that the organisation's goals under Warchus' leadership include the continuation of outreach work, the establishment of an endowment fund and to "oversee redevelopment of the building following the bicentenary in 2018."
Spacey, who is soon to appear in a one man production at the Old Vic, Clarence Darrow, added: "I know he loves our theatre and I am delighted for our staff, our audiences and for our acting and production communities that he will be its next guardian."
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