A child killer and rapist has been executed in Ohio after apologising to the three-year-old girl's family, praying and shedding a tear.

Ronald Phillips, 43, was condemned to die for the 1993 rape and slaying of his girlfriend's toddler daughter Sheila Marie Evans in Akron.

As he lay on the execution table at the Southern Ohio Correctional Facility, he apologised to the child's aunt and half-sister, who were there to watch him die for his crimes.

"I'm sorry you had to live so long with my evil actions. All those years I prayed you'd forgive me and find it in your heart to forgive and have mercy on me," he said.

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"Sheila Marie did not deserve what I did to her. I know she is with the Lord and she suffers no more. I'm sorry to each and every one of you that you lived with this pain all those years."

Donna Hudson, the victim's aunt, said: "God forgave him, but, I'm sorry, I don't think I can."

He appeared to pray and shed a tear before he died. He died about 10 minutes after giving his final statement. He showed no signs of distress. His chin dropped and his belly heaved slightly as the lethal drugs were administered.

Phillips severely beat Sheila in the head, face and abdomen, threw her against walls and dragged her by the hair. He also sexually assaulted her.

Phillips' last meal on Tuesday consisted of a large cheese pizza with bell peppers and mushrooms, a two-litre bottle of Pepsi and strawberry cheesecake, along with grape juice and a piece of unleavened bread.

Phillips lost his final appeal on Tuesday when the U.S. Supreme Court denied his requests for more time to pursue the challenge to the new drug combination or his claim that he deserved mercy because he was only 19 at the time of the crime.

Phillips staved off lethal injection three other times, including in 2013 when he made a last-minute request to donate a kidney to his mother and possibly his heart to his sister.

The request was ultimately denied. His mother has since died.

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His execution on Wednesday was Ohio's first since a problem-plagued one three-and-a-half years ago triggered an uproar over the reliability of the lethal injection drugs used by the state.

His case could open the way for the full resumption of capital punishment in Ohio, which has 26 executions scheduled through 2020.

"I have confidence that we are going to continue to do this in a dignified, peaceful, humane way, and I'm committed to do that," Ohio Prisons Director Gary Mohr said just before Phillips' execution.

In the 2014 incident, inmate Dennis McGuire gasped and snorted repeatedly during a procedure that took an unusually long 26 minutes and involved a never-before-tried drug combination.

Gov. John Kasich reacted by putting all executions on hold. The delays continued when the state had trouble finding new supplies of drugs and death row inmates sued over Ohio's proposed new three-drug combination, saying it would amount to cruel and unusual punishment.

The drugs include midazolam, a sedative used in some problematic executions in Ohio, Arkansas and Arizona. The inmates were backed up by 15 pharmacology professors who said midazolam is incapable of inducing unconsciousness or preventing serious pain.

Allen Bohnert, a public defender who worked on Phillips' case, contended his execution was not problem-free, it just looked that way. He said the executioners accelerated administration of a paralytic to mask Phillips' pain.

"Hiding the physical evidence does not change the reality that Ohio used a painful and unnecessary method of execution to kill Ron Phillips today," Bohnert said.

It was the nation's 15th execution of the year.