This issue has been traced to a single battery supplier that produced batteries with a manufacturing defect. While the possibility of an incident is rare, the likelihood increases as the battery ages.
Apple recommends that you stop using your iPod nano (1st gen) and follow the process on their website to order a replacement unit, free of charge.
Note: This battery issue is specific to the iPod nano (1st gen) and does not affect any other iPod.
(Thanks Jonathon)