Ferrari had announced his departure on Monday, with Cardile likely to have an extended period of gardening leave.
"Enrico shares my motivation to be successful in F1 and will have all the resources available to him to realise that ambition," team owner and CEO Lawrence Stroll said in a statement.
"Together with Andy Cowell joining as Group CEO in October and our existing leaders we are creating a formidable team."
Former Mercedes engine boss Cowell was announced last week to take the place of Martin Whitmarsh, the former McLaren principal.
Formula 1 faces major change in 2026 with a new engine and technical regulations.
Aston Martin will be switching from their current Mercedes power units to become the only team supplied by Honda.