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Minghella, who won the Best Director Oscar for 1996 film The English Patient, was helming one of a collection of 12 short love stories under the title New York, I Love You.
He asked Kapur to take over before he went into hospital for surgery for cancer of the tonsils and neck earlier this month. The 54-year-old suffered a fatal haemorrhage last Tuesday at London's Charing Cross Hospital.
Scrubs star Zach Braff, Brett Ratner, Mira Nair and Scarlett Johansson are also rumoured to be directing segments of the Woody Allen film.
Kapur said: "Anthony, who was doing one, called me one day and said, 'I'm not doing it. I'm ill. You do it.' Anthony went to the operating table, but did not return. He was 54 and it was not an age to die."
Kapur's previous credits include Elizabeth and sequel Elizabeth: The Golden Age.
> Minghella premiere was 'bitter-sweet'
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