The 23-year-old moves to the iPro stadium on a four-year deal and will link up with former Villa striker Darren Bent.
The Austrian joined Villa as a 15-year-old from Rapid Vienna and made 134 appearances, scoring 21 goals.
Weimann, who has 13 international caps, became a peripheral figure under Villa manager Tim Sherwood.
He is Derby's fourth signing of the summer following the arrivals of  Bent, Alex Pearce and Scott Carson.
Manager Paul Clement said: "Andreas is an exciting talent and he will be a positive addition to the squad.
"He has a good amount of top-flight experience to his name and, at the age of 23, he still has a lot of potential."
Weimann's most prolific season was 2012/13 when he scored 13 times and in 2014 he ensured Villa's Premier League status with a brace against Hull in the final home game of the season.
"The opportunity to play first-team football and drive the team to promotion is something that I want to do," added Weimann.
"Derby has plans to be successful and get promoted and that's what attracted me here; I want to be part of that."
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Derby County have signed forward Andreas Weimann from Aston Villa for an undisclosed fee.