Robert Ladd, who had an IQ of 67, was executed by lethal injection at 7:02 pm (0102 GMT today) in the state's death chamber in Huntsville, justice department spokesman Jason Clark said.
Ladd's lawyers had asked the Supreme Court to intervene because some mentally impaired death-row convicts have been spared. A second appeal based on the lethal injection was also denied.
After mentioning several people by name and thanking his family, Ladd's final words were, "Let's ride."
The court cited the slow-minded character Lennie from John Steinbecks's "Of Mice and Men" as an example of someone who "might" be exempted from the death penalty.
"This case is indeed stranger than fiction. Anywhere else in the country, Mr Ladd's IQ of 67 would have meant a life sentence, not death," said Ladd's attorney Brian Stull, of the American Civil Liberties Union's capital punishment project.
He said Texas had severely misjudged Ladd's mental capacity and was "relying on standards for gauging intelligence crafted from 'Of Mice and Men' and other sources that have nothing to do with science or medicine."
The lawyer added that Ladd, 57, was "fairly obviously retarded" and his life had been "full of evidence of his intellectual disability."
A Texas psychiatrist had deemed Ladd mentally impaired from the age of 13 and by 18 he was in a specialist center for people with mental handicaps, according to a court document.
Maurie Levin, another of Ladd's attorneys, had also tried to get the execution halted by calling into question the quality of one of the drugs used to kill him, pentobarbital.
Ladd was sentenced to death for the 1996 rape and murder of a young woman whose apartment he had raided and burned.
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