The production, which is directed by Paul Miller, ran as part of a dedicated Michael Frayn season in the Crucible Theatre last month. The Old Vic run marks the first London transfer under Daniel Evans' tenure as Sheffield Theatres' artistic director.
Evans said he was "thrilled" at news of the transfer, adding: "We are proud to produce high quality work which attracts audiences from the city, the region and beyond, and it is wonderful that this opportunity will allow us to bring our work in Sheffield to an even wider audience."
Kevin Spacey, artistic director of the Old Vic, said: "I am delighted that we are continuing our relationship with Michael Frayn following the great experience we've had together with Noises Off. Michael's work is compelling and we're excited now to work with Daniel Evans and Sheffield Theatres to bring their terrific production of Democracy to a wider audience."
The 2003 play is about the geopolitical ambitions of an inspirational politician and the forces in play around him, providing a timely exploration of the Machiavellian nature of coalition government.
West German Chancellor Willy Brandt - one of the most charismatic leaders in post-war politics - makes history as he starts to reunite Europe; but he discovers things aren’t as straightforward as they seem when a spy is uncovered in his office and his plans are thrown into chaos.
The cast features: Andrew Bridgmont, David Cann, Patrick Drury, Richard Hope, William Hoyland, Ed Hughes, David Mallinson, Aidan McArdle, James Quinn and Rupert Vansittart.
Michael Frayn said: “All three of the productions that Sheffield Theatres did of my plays earlier this year were terrific. I’m so pleased that at any rate one of them ... has found a London home - if only because I so much want to see it again myself. And a particular pleasure to have it at the Old Vic, still warm from their Noises Off.”