"FIFA and the Local Organising Committee (LOC) learnt of the death of the worker on Saturday at the Arena Amazonia site with great sadness," the statement read.
"We would like to send our most sincere condolences to his family, relatives, colleagues and friends."
Marcleudo de Melo Ferreira's death highlighted the safety concerns surrounding Brazil's ability to host the 2014 World Cup as he was the fifth worker to die while building Brazil's 12 World Cup stadiums.
Ferreira died in hospital after a broken cable led to him falling nearly 35 meters (115 feet).
Fatal accidents have occurred in Manaus, Brasilia and most recently in Sao Paulo, where two people died on November 27 after a crane collapsed in the arena that is to host the opening game on June 12.
Another worker died while working on Palmeiras arena in Sao Paulo, but that stadium will not be used in the World Cup.
The tournament will take place in 12 Brazilian cities.
Preparations for the competition - the first to be held in South America since 1978 - have been plagued by delays, accidents, cost overruns, and public anger over government waste that contributed to massive nationwide street protests last year.
The third death in less than a month again raised concerns that safety of workers was being ignored as officials rush to finish their arenas. Six stadiums are due to be delivered between now and April, and several are well behind schedule.
FIFA president Sepp Blatter last week appealed to "God, Allah and whoever" to ensure the stadiums will be ready in time.
Andrade Gutierrez, the Brazilian firm building the Amazonia stadium, said in a statement that Ferreira worked for a company that had been contracted to build the arena's cover and an internal investigation of the incident would be conducted.
Manaus will host four games, including the high-profile encounter between former World Cup winner England and Italy.
(Reporting by Bruno Kelly; Writing by Caroline Stauffer; Editing by Doina Chiacu and Pritha Sarkar)
