The 53-year-old, who was an Australian Under-21 international and played for French club Racing Metro, will start his role on August 15, the IRB announced.
He will be responsible for day-to-day IRB operations and have the dual role of Rugby World Cup managing director.
Gosper said: "I am delighted to be taking on this role at what is such an exciting time for the game.
"Rugby World Cup going is from strength to strength as a major global sporting brand as we look forward to the biggest-ever Rugby World Cup in 2015 (England) and the sport returning to the Olympic Games at Rio 2016.
"I am passionate about rugby and look forward to using the experience gained through my involvement with the sport and management of global organisations and brands to work with the IRB Council, in partnership with unions and regions."
IRB chairman Bernard Lapasset said the chief executive selection had been made after "an extensive recruitment process".
"Brett Gosper proved himself to be an outstanding candidate during an extensive recruitment process featuring excellent candidates," Lapasset added.
"The IRB executive committee unanimously agreed that his strong blend of experience, skills and passion for the game are perfectly suited to the role.
"Brett joins a strong and dynamic management team at an exciting time for the IRB and rugby, and I am confident he will continue to drive the game forward."
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