Olivia Colman was previously named as the replacement for Claire Foy as Queen Elizabeth II in the Netflix series.
London-born Menzies, 44, played Edmure Tully in Game Of Thrones and is also known for his Golden Globe-nominated performance in Outlander.
Matt Smith starred opposite Claire Foy for two seasons (Ian West/PA)
The announcement came after the series suffered a bout of bad publicity over its gender pay gap when it was revealed Smith was paid more than Foy, who won a Golden Globe for her performance.
Production company Left Bank Pictures apologised to both stars and said they were not aware of how much each were being paid.
