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Rhodri Philipps, 50, the 4th Viscount St Davids, made the threat in November, just days after Miller won a legal case requiring the government to consult parliament before beginning Britain's withdrawal from the European Union.
"£5,000 (5,600 euros, $6,400) for the first person to 'accidentally' run over this bloody troublesome first generation immigrant," he wrote.
He described Miller, who was born in what was then British Guiana, as a "boat jumper" and added: "If this is what we should expect from immigrants, send them back to their stinking jungles."
Senior district judge Emma Arbuthnot said the comments were racially aggravated and warned Philipps -- who is also known as Lord St Davids -- faces jail when he is sentenced on Thursday.
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In a statement read to Westminster Magistrates' Court in London on Monday, Miller said the comments left her "very scared for the safety of herself and her family".
Prosecuting lawyer Philip Stott said: "She took the threat seriously, and it contributed to her employing professional security for her protection."
Philipps, who lives in the exclusive London district of Knightsbridge, accepted writing the posts but said they were not menacing and were not publicly visible. Prosecutors said they were.
Miller was subjected to torrents of abuse for her legal challenge, which critics took as an attempt to block Brexit. The government began the process of leaving the EU in March.