Carter tried to mend ties with Cuba while in office
But he said the administration did suspect Fidel Castro's government of possessing the capability to conduct biological research for offensive purposes.
Mr Powell was speaking after the issue arose during Jimmy Carter's landmark goodwill visit to the country, which fervently denies having any such programme.
On a visit to a bio-technology laboratory in Havana, the former president said there was no evidence of Cuba exporting technology that could be used for "terrorist" purposes.
But Mr Powell, speaking on his way to the Nato-Russia talks in Iceland, said: "We do believe Cuba has a biological offensive research capability."
"We didn't say it actually had some weapons, but it has the capacity and capability to conduct such research."
On Friday, Cuban President Fidel Castro accused the US of lying after it added his country to a list of states suspected of trying to acquire weapons of mass destruction.
US Undersecretary of State John Bolton had said Cuba was believed to possess a "limited offensive biological warfare research and development effort" and warned it could become a target in the US-led war against terrorism.
Mr Carter said on Monday that he had consulted senior US officials before his visit and was told there was no evidence that Cuba had been involved in sharing information that could be used for terrorism.
The former Democrat president, who has been involved in peace work since leaving office in 1981, said that an "allegation of bioterrorism" had been made "maybe not coincidentally" just before his visit to Cuba.
Human rights
The plant which Mr Carter toured on Monday is, according to Cuba, designed to produce medicines and vaccines with a view to exports.
Cuba is also involved in developing a similar plant at Karaj in Iran.
But he made it clear that there were irreconcilable differences between the US and the Cuban Communist leadership on the issues of democracy and human rights.
"It is very important to have absolute freedom of expression and freedom of assembly," he told Cuban students, adding that people in the United States greatly valued their freedom to criticise their government and replace it in free elections.
Mr Carter has met veteran civil rights activists Elizardo Sanchez and Oswaldo Paya and plans to meet the country's highest-profile dissident, Valdimiro Roca, who was released last week after nearly five years in prison.
Later on Tuesday, he is scheduled to make an unprecedented address to the nation which, Mr Castro has promised, will be broadcast live on state media.
Mr Carter, whose visit ends on Friday, is the most prominent American to visit the island since Fidel Castro took power in the 1959 revolution.
He has been warmly welcomed by Mr Castro for his opposition to US sanctions and attempts to achieve a rapprochement between the two nations during his term in office from 1976 to 1981.