Chief executive Martin Long - who has been with Game for 13 years - will step down. Game chairman Peter Lewis said: 'Now is the right time to make a change'.
The company said that Long was leaving the business in a strong trading position and with excellent future prospects.
Morgan had been deputy chief executive at Game since 2004 and has 'unrivalled experience in the video games industry,' said Lewis.
He added that the new top team meant that Game was now 'well placed to deliver on UK and international growth plans'.
Lewis also thanked Long for helping Game to grow its empire from 16 stores to almost 800 in eight countries, making it the clear market leader in Europe.
Shares in Game eased down slightly on the news to 87.75p - a fall of less than 1 per cent.