"This is great for basketball back home. Basketball has been getting better and better and this is another step to where we want to get to."
Basketball is the only Olympic sport where a decision on host participation has to be approved by the world governing body.
Had GB's teams not been granted host spots, they would have had to reach the finals of their respective EuroBasket competitions.
GB women only qualified for the tournament for the first time this year.
Leedham and the GB women's team will play at EuroBasket this summer
The men's team has made significant progress since they were reformed by the individual federations of England, Wales and Scotland in 2005.
They reached EuroBasket in 2009 and 2011 and, although they failed to get out of the group stages two years ago, have been tipped as an outside contender for a medal at 2012.
Fiba had insisted that GB produce a legacy plan for the sport post-2012 to match their accomplishments on the court.
"The key to today was the team's results last summer, when they qualified for EuroBasket as of right," said British Basketball Federation chairman Bill McInnes. "That made it a very easy decision for Fiba to make.
"But five years ago people thought we were out of our heads to dream of playing in the Olympics."
The Fiba board's decision was taken by a 17-3 majority on a proposal that the have to decide before next July whether the three federations want to carry on as Great Britain or devolve back to individual nations for competitive matches.
Originally, GB asked Fiba for four years to make that decision and transition, but the decision gives them only 15 months to decide their post-2012 future.
Fiba's desire for a unified Great Britain body within Fiba is a concept that has met with some opposition domestically.
We are confident the British players will inspire new generations of basketball players across the UK
Andy Hunt BOA chief executive
Fiba secretary general Patrick Baumann said the vote was a sign that the governing body wanted GB to show decisiveness in determining their future.
"It's not about losing patience," he said. "I think they are so close to making the right steps for the right governance and really driving basketball for the future.
"It's in everyone's best interests - I understand the sensitivities in the country and so we left them the choice"
GB Basketball has implemented a plan called Towards 2016, designed to develop the game domestically as well as ensure the teams qualify in their own right for the next Olympic Games in Rio de Janeiro.
It has also commissioned a feasibility study into hosting the Men's EuroBasket tournament in either 2015 or 2017 as another sign of long-term commitment.
The news means Great Britain are the second nation to book their place in next year's Games, after the US men's and women's teams won their respective world championships last year.
British Olympic Association chief executive Andy Hunt welcomed the announcement.
"We are delighted that Fiba has rightfully given both the teams the opportunity to compete for Team GB in front of an enthusiastic home crowd," he said.
"Today's decision is recognition of the excellent progress made by British Basketball on and off the court over the last five years.
"It represents a once-in-a-lifetime opportunity for the sport to significantly increase its participation and profile, and leave a lasting legacy throughout the UK.
"We are confident that the British players will thrill the public with credible, exciting performances and by showcasing their skill and talent, inspire new generations of basketball players across the UK."
NBA commissioner David Stern welcomed the decision, saying: "The opportunity for [British] fans to see Team GB compete at the highest level of our sport will be a treat for them and will accelerate the appreciation for basketball in the UK in an enormously positive way."
GB gave eventual winners Spain a scare at EuroBasket 2009
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