Texas has executed a child killer at the centre of a bitter dispute over what defines legal mental impairment after several courts ruled that he could be put to death despite a low IQ.

Bobby Wayne Woods, 44, was killed by lethal injection for raping and murdering his girlfriend's 11-year-old daughter after the US supreme court denied a last minute appeal by lawyers who argued that the condemned man fell within a ruling by the same court in 2002 that the mentally impaired could not be sentenced to death.

After being told that the supreme court had refused to intervene, Woods' last words were: "Bye. I'm ready."

Tests on Woods in prison put his IQ as low as 68, below the widely accepted cut off for mental impairment of 70. However, in pursuing the death penalty, the state fell back on other tests conducted when he was a child that put his IQ as high as 86.

Maurie Levin, a University of Texas law professor who represented Woods, said that he is "transparently childlike and simple" and described the execution as "a travesty".

Despite the supreme court ruling seven years ago barring the death penalty for the mentally impaired, individual states are left to decide what defines severe learning disabilities. The court said that an IQ test of "around 70" was an indicator but did not set a specific test.

Recent studies have shown that men with IQ scores lower than 70 have been executed in several states, including Texas and Alabama.

Earlier this week, the Texas board of pardons and paroles, voted unanimously that Woods, who has been described as barely literate, should be executed. The Texas attorney general, Greg Abbott, argued to the supreme court that the IQ tests on Woods were unreliable.

"The only experts to ever conclude that Woods was mentally retarded did so after he had committed this murder and had motivation to underperform," he said.

Richard Hattox, who prosecuted Woods, defended the execution.

"Woods testified, and the jury watched him reason and think and debate with me and his own lawyers," he said. "If he was going to claim mental retardation, having him testify was a mistake, because that took away any doubt."

Woods raped and slit the throat of Sarah Patterson after abducting her with her brother, Cody. The boy was badly beaten but survived.

The children's mother, Schwana Patterson, was refused permission to watch the execution because she served a prison sentence for failing to protect her daughter from sexual abuse by Woods.

Woods was the 24th person executed by Texas this year.