Kelly Gissendaner, 47, was the first woman put to death in the southern US state in 70 years.
Lawyers filed at least three appeals with the US Supreme Court to try to delay the sentence hours before she died.
Gissendaner planned but did not carry out her husband’s murder in 1997.
Her former lover, Gregory Owen, who killed Douglas Gissendaner, was given life in prison as part of a plea bargain.
Pope Francis, who was recently on a US tour, urged the review board to reconsider. But on Tuesday afternoon, the board announced it was not granting clemency.
Hours later, the US Supreme Court said it had rejected three applications for a stay of execution.
Witnesses to the execution told local media that she sang Amazing Grace before being given a lethal injection.
“Bless you all,” she was quoted as saying by the Atlanta Journal-Constitution. “Tell the Gissendaners I am so, so sorry that an amazing man lost his life because of me. If I could take it all back, I would.”
The Pope’s appeal for Gissendaner’s life was made in a letter written by his diplomatic representative in the US, Archbishop Carlo Maria Vigano, on Tuesday.
The archbishop wrote that, while not wishing to minimise the gravity of the crime, he implored the board “to commute the sentence to one that would better express both justice and mercy”.
Gissendaner’s lawyers told the board she had undergone a transformation in prison, offering support to troubled inmates and showing remorse for her own crime.
Former Georgia Supreme Court Chief Justice Norman Fletcher also made representations to say Gissendaner’s death sentence was disproportionate to the crime that she committed.
He noted that Georgia had not executed a person who had not committed the actual killing since the death penalty was reinstated in 1976, and said a court process to review an appeal by Gissendaner had been “deeply flawed”.
Georgia has executed nearly 60 people since 1976, and has more than 80 people on death row.
Gissendaner’s former lover Owen will become eligible for parole in 2022. He took a plea deal and testified against Gissendaner, and for his co-operation was sentenced to life in prison.
Douglas Gissendaner’s family said in a statement on Monday that Kelly Gissendaner’s sentence was appropriate.
“She had no mercy, gave him no rights, no choices, nor the opportunity to live his life,” the family said.
Gissendaner has had two previous execution dates.
Her execution was rescheduled in February because of a winter storm that was forecast to hit Georgia, and the next date in March was cancelled after officials said the drug used in the lethal injection was cloudy.