Letterman has been a late-night fixture for 20 years

Letterman, 55, was absent from his legendary CBS chat show in the US on Wednesday night.

Willis had already got the Late Show in his diary, but had been due to appear as a guest.

I look like somebody gave me a beating David Letterman on his eye infection

It was the first time Letterman had taken time off from his hit show since undergoing open-heart surgery in early 2000.

He had to miss several weeks of shows and a series of guest hosts filled in until Letterman was able to return full-time in March 2000.

Before then, Letterman, 55, had never missed an appearance in more than 20 years of working on late-night television.

Letterman's production company issued a statement from the comedian's doctor.

"Dave has an infection of his right eye. It's under treatment and we expect a full recovery," said Dr Louis Argonne.

Willis was due on as a guest

As stand-in, Hollywood star Willis played host to actress Carmen Electra, singer John Mayer and CBS Evening News anchor Dan Rather.

Rather has just returned from filming an exclusive interview with Saddam Hussein. It was the Iraqi leader's first interview with a US TV journalist for 12 years.