The position has been vacant since the death of co-founder Steve Jobs last month.
Levinson is the chairman of Genentech, as well as being a co-lead director of Apple's board since 2005. He joined the company in 2000. He also served as a director of Google, Inc. from 2004 to 2009.
Apple has also assigned Disney's Robert Iger to its board of directors.
Of Levinson's appointment, Apple's chief executive Tim Cook praised his "insight and leadership", and said that he is a great fit for Apple.
"Art has made enormous contributions to Apple since he joined the board in 2000," Cook said. "He has been our longest serving co-lead director, and his insight and leadership are incredibly valuable to Apple, our employees and our shareholders."
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Levinson added: "I am honoured to be named chairman of Apple's board and welcome Bob to our team.
"Apple is always focused on out-innovating itself through the delivery of truly innovative products that simplify and improve our lives, and that is something I am very proud to be a part of."
Iger said: "I am extremely pleased to join the board of such a wonderful company. Over the years, I have come to know and admire the management team, now ably led by Tim Cook, and I am confident they have the leadership and vision to ensure Apple's continued momentum and success."
The company's shares rose 2.52% at $388.83 (£245.83) shortly after the news was announced.
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Scroll through a gallery of images of the late Apple chairman Steve Jobs's career below: