Apple refused to admit any problems with the small and sleek units after they were launched earlier this month, but an increasing number of complaints about the LCD screens forced today's climbdown.
One disgruntled Nano owner, Matthew Peterson, was so frustrated with the problems that he set up the website flawedmusicplayer.com, on which he invited other Nano owners to share their experiences and posted pictures of cracked screens.
The Nano has proved so popular that the high street retailer Dixons is reportedly selling five every minute.
However, an investigation by BBC's Watchdog programme revealed that customers were not being told that the rechargeable batteries used in most of the portable music players had a limited life span.
Last night, Apple acknowledged there was a problem with one particular shipment of Nanos, offering customers either a full refund or a replacement.
However, the company has not disclosed how many of the units have been affected, saying it was only "a tiny fraction" of the amount already sold.
"The iPod Nano screen-cracking problem is a real but minor issue involving a vendor quality problem in a small number of units," Phil Schiller, an Apple spokesman, said.