Scotland Yard said 30-year-old Ashraf Islam pleaded guilty to the offence at Uxbridge Magistrates' Court on May 25 and is being held in custody awaiting sentencing.
He was arrested after he walked into the Hounslow police station in west London on May 23, a day after the murder of soldier Drummer Lee Rigby on the streets of Woolwich.
Officers from the Metropolitan Police's Counter Terrorism Command investigated the case, but no terror charges were brought against Islam, a spokesman for Scotland Yard said.
He is believed to have become a Muslim during a recent spell in prison and now faces up to 10 years in jail for the death threats.
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Scotland Yard allegedly also found a laptop belonging to him showing internet searches for kidnapping, guns and vans.
His internet history is also said to show he had been on terrorist and firearms websites.
According to the Sunday Mirror, a review of Prince Harry's security was launched in the wake of Lee Rigby's murder as it is feared that the 28-year-old might be a prime target for the Taliban after touring Afghanistan twice with the Household Cavalry in 2008 and the Army Air Corps last year.
The prince is also said to have been given extra security by Britain's elite Special Air Service.