Final answer:
Emperor Nero is said to have ordered the execution of the Apostle Paul, especially in the context of the early Christian persecution that began around AD 64.
Step-by-step explanation:
The Roman emperor who supposedly ordered the execution of the Apostle Paul was Nero. According to Church tradition and historical records by the Roman historian Tacitus, it was during Nero's reign, specifically around the time shortly after the Great Fire of Rome in AD 64, that both Peter and Paul were martyred in Rome.
The official persecution of Christians is recorded to have begun under Nero's rule, making him a notable emperor in the context of early Christian history.