Update: Thanks to tipster Stefan for digging into Nintendo's bowels and unearthing the replacement request form. Good news, looks like they'll simply ship the new strap without the mess of having to return the old one.
Update 2: Nintendo just called to let us know that the replacement program is entirely voluntary, and does not constitute a "recall". That means Nintendo apparently doesn't think it's a risk for most people, but if you do you can feel free to get that Wiimote strap. Nintendo issued this statement: "Nintendo has a long tradition of delivering high-quality products and excellent customer service. The Wii Remote is meant to be gripped, and the wrist straps are a secondary measure. We have found that some consumers get more excited playing Wii Sports than we could have imagined. This resulted in a very small number of wrist straps breaking. We have replaced the broken wrist straps and continue to reinforce with consumers the safe and fun use of our products. Through its voluntary wrist strap replacement program Nintendo is offering a stronger wrist strap free of charge to consumers by calling 1-800-859-4519 or by visiting support.nintendo.com."
Update 3: Here's the info Nintendo gives when you order your replacements: "Thank you for your request. Once your replacement wrist strap order has shipped, you will receive a confirmation email from Nintendo. We expect to begin shipping replacement straps around December 21st. It will take 5 to 9 days for delivery depending on your location. Please do not contact Nintendo regarding your replacement wrist strap until after that time period has passed."
[Thanks, Matthew P.]