Frew, 63, is currently chair of FTSE 100 chemicals group Croda and a former deputy chair of Lloyds Banking PLC. She will join the Rolls’ board next month and take up her new position at the start of October.
The Scottish businesswoman succeeds Sir Ian Davis, who has been chairing the engineer for almost nine years.
Sir Kevin Smith, Rolls’ Senior Independent Director, said: “Anita is the unanimous and clear choice of the Nominations & Governance Committee and her proposed appointment has been warmly endorsed by the entire board.”
Rolls-Royce has endured a traumatic year with the company badly affected by the restrictions on travel imposed during the Coronavirus (COVID-19) lockdowns.
The engineer posted a loss of nearly GBP4bn in 2020, cut 7,000 jobs and raised GBP7.3bn through loans, export credits and a GBP2bn share issue to keep going.
In May, it said hoped to start generating cash again in the second half of 2021.
Warren East, chief executive, added: “I would like to thank Ian for his steadfast leadership, guidance and unwavering support during a crucial period for the group, both strategically and operationally. Ian has played a key role in contributing to Rolls-Royce’s future.
“I am delighted to welcome Anita to Rolls-Royce, with her extensive experience and leadership I look forward to working with her.”
Rolls also announced the appointment of Mike Manley to the board from 1 July.
“He is currently head of Americas of Stellantis and a member of their executive team,” said the statement from Rolls.