A 51-year-old British grandmother could be killed by lethal injection in less than three months unless her family and campaigners succeed in a last-ditch attempt to gain clemency from the governor of Texas.

Linda Carty, a drug enforcement agency undercover agent, has spent the last eight years on death row after being convicted of planning a macabre murder.

The supreme court has now refused to reopen Carty's case, which means all legal avenues have been exhausted. Execution dates are generally set from one to six months from this stage, with two to three months being typical.

Carty, who was born on the Caribbean island of St Kitts when it was a British colony, was found guilty of hiring four men to murder her 25-year-old neighbour, Joana Rodriguez, so that Carty could cut the baby out of Rodriguez's womb and claim the child as her own.

That Rodriguez had given birth four days earlier is just one of the facts that make the prosecution's case unlikely, her supporters say. Carty would also have found it impossible to pass off the baby – later found safe – as her own: Rodriguez was of a different ethnicity.

Carty's daughter, Jovelle Carty Joubert, is appealing to the governor of Texas for clemency. "It is completely impossible that my mother committed these crimes," she said as she flew into the UK today for a meeting with the Foreign Office to ask for help to save her mother's life. "But she had no chance to clear her name. Her trial was so unfair. We are trapped in a legal system where if you don't have money, you don't have rights or justice."

Clive Stafford Smith, the founder and director of Reprieve who is fighting for Carty, said the case was catastrophically flawed and the "most desperate, outrageous miscarriage of justice case we've yet faced in 2010".

"It's urgent now," he said. "Texas is efficient at executing people – 99.6% of cases that get to this point go through to the end. We're asking the British government to not hold back. After all, there's no limit to what the government should not do for a citizen about to be killed by a foreign country."