Senegal midfielder Diame, 29, joins for an undisclosed fee on a three-year deal after rejecting a new contract at Hull.
Clark, 26, has signed for a £5.1m fee on a five-year contract, having been at Villa since he was 11.
He was only 12 months into a five-year deal at Villa, who were relegated to the Championship with Newcastle in May.
"Ciaran is a very versatile player," said Magpies manager Rafa Benitez. "As a left-footed centre-back, he can also play left-back and as a defensive midfielder."
Clark was part of the Republic's Euro 2016 squad and has scored 10 goals in 159 Villa appearances since making his debut in August 2009.
Former Wigan Athletic and West Ham United player Diame scored 10 goals in 45 appearances for Hull last season, including the winning goal as they beat Sheffield Wednesday 1-0 in the play-off final at Wembley.
"Everybody close to me knows how desperate I am to play in the Premier League again, but when Newcastle comes in for you it makes you think," he told Newcastle's club website.
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Newcastle United have signed Mohamed Diame from Hull City and defender Ciaran Clark from Aston Villa, having activated release clauses for both.