South Africa Rugby Union (SARU) Chairman Oregan Hoskins was elected IRB Vice Chairman.
Hoskins, who up against Beaumont for the role, was elected after securing the Chairman's casting vote following two rounds of voting which were tied 13-13.
Lapasset and Hoskins will begin their term in office on January 1, 2012.
Lapasset has presided over rugby's return to the Olympic Games, Argentina's inclusion in the Rugby Championship, a 10-year tours and Test schedule, the awarding of the Rugby World Cup to an Asian country for the first time and an exceptional 2011 Rugby World Cup during his recent term.
Lapasset thanked Council for its continued support, paid tribute to outgoing Vice Chairman Beaumont and outlined his vision for rugby's world governing body.
"I am honoured to accept the mandate of the Council to serve as Chairman of the International Rugby Board for another term," he said.
"Together we must work to ensure that our sport remains strong for all our Unions and is able to continue its phenomenal growth around the world.
"I am committed to building on the good progress to date. We have fantastic opportunities to grow the game, to reach new markets and welcome new members to the rugby family.
"Together we also have to address the issues that face our sport and many other sports, including the delivery of a robust financial model for sustainable growth for our Unions, upholding our values, promoting the best possible welfare for our athletes and maintaining integrity."
Hoskins commented: "It is an honour to accept this mandate. I am looking forward to working in partnership with Bernard and my colleagues on Council to ensure that rugby is best placed to thrive as a sport at both the elite and community levels as we continue to build the platform for sustainable growth in the decade ahead that will include two Rugby World Cups and Rugby Sevens' Olympic Games debut."
The original meeting in Auckland, New Zealand, on October 19 was adjourned in order to defer the vote of the Chairman and Vice Chairman positions until after the Rugby World Cup.
As per the original meeting agenda, the Council also elected the seven representatives to serve on the IRB Executive Committee, alongside Chairman Lapasset, Vice Chairman Hoskins and Chief Executive Mike Miller, for four years from January 1, 2012.
They are: Bill Beaumont (England), Tatsuzo Yabe (Japan), Giancarlo Dondi (Italy), Peter McGrath (Australia), Peter Boyle (Ireland), Graham Mourie (New Zealand) and Bob Latham (USA).
It is the first time that Japan and USA have been represented on the IRB Executive Committee and the first time that two Tier 2 Unions have been represented.
The current mandates of Lapasset as Chairman and Beaumont as Vice Chairman and the Executive Committee expire on January 1, 2012.