The arch-Brexiteer was also appointed Lord President of the Council and will attend cabinet, Downing Street said.
Mr Rees-Mogg, chairman of the European Research Group, told Sky News: “The prime minister kindly offered me a very interesting job to do, one that is something that I’m very interested in because parliamentary procedure and practice is something I’ve spent a lot of time on.”
He described the new prime minister as “incredibly businesslike and ready to crack on and deliver for the country”, adding: “It’s a man with a mission and that mission is to get us out of the EU by 31 October.”
Asked about the appointment of Vote Leave campaign director Dominic Cummings as an adviser to Mr Johnson, Mr Rees-Mogg said: “Dominic is an incredible intellect, a very effective administrator, look what he did at education with Michael Gove, so I think this viewing of it purely as an election campaigning matter is not necessarily accurate. Yes he did brilliantly for Vote Leave and Vote Leave won, but Dominic has been hugely successful in the nitty gritty of policy.”
Mr Rees-Mogg added: “The parliamentary arithmetic doesn’t change. But the thing to bear in mind is that the Article 50 act and the Withdrawal Act mean that the default position is we leave on 31 October.
“And that would have to be changed to stop the law taking its course. So it’s not simply a matter of a parliamentary motion, parliament would have to change the law and it’s hard to see how that would happen.
“The House of Commons will run in its normal way and it will be my role as Leader of the House to ensure that parliamentary business is the business parliament ought to have in accordance with our constitutional norms, which I attach great importance to.”
Mr Johnson launched a full-scale reshaping of Ms May’s government on his first day as prime minister.
Dominic Raab was appointed foreign secretary and first secretary of state – effectively making him Mr Johnson’s deputy prime minister.
The new cabinet also sees Michael Gove, Mr Johnson’s Vote Leave colleague who scuppered his last leadership bid, moved to become the Chancellor of the Duchy of Lancaster and replaced as environment secretary by fellow Brexiteer Theresa Villiers.
Gavin Williamson, who was sacked from his defence role three months ago over suspicions he leaked details of Huawei discussions from the National Security Council, has become education secretary.
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