"It is with [a] heavy heart I share that my wonderful brother, Robert, peacefully passed away tonight," he said in a statement on Saturday.
The president had visited his brother in hospital in New York the day before his death, telling reporters: "He's having a hard time."
It is unclear what caused Robert Trump's death.
A number of US media reports suggested he had been seriously ill.
"He was not just my brother, he was my best friend," Donald Trump said in a statement on Saturday. "His memory will live on in my heart forever."
President Trump's son Eric described his uncle as an "incredible man - strong, kind and loyal to the core".
"He will be deeply missed by our entire family," he wrote on Twitter.
Trump's parents and siblings: What do we know of them?
Robert was the youngest of Fred and Mary Anne Trump's five children, and was born two years after his brother D
Robert Trump spent much of his career with the family real-estate firm, becoming a top executive. Unlike his brother, however, he was said not to court publicity and lived semi-retired in New York state.
He recently went to court in a failed bid to stop publication of his niece Mary Trump's tell-all book about the president, How My Family Created the World's Most Dangerous Man.
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The eldest of the children, Fred Jr, died in 1981.