Apple's MagSafe adapters are apparently so faulty that the company has decided to replace all of them, free of charge, whether your MacBook is in warranty or not. Just make an appointment at the Genius Bar (at your local Apple Retail Store) and they'll set you up with a new MagSafe adapter in no time.

From Apple's Support page:

Symptoms

You may notice a separation of the white insulation on the magnetic end of the MagSafe adapter. Under continued use, the cable may discolor and the rubber molding may become deformed.

Products Affected

MacBook (13-inch Late 2006), MacBook (13-inch), MacBook Pro (15-inch Glossy), MacBook Pro (17-inch)

Resolution

Whether your product is in or out-of-warranty, you can take your adapter (you don't need to take the computer) to an Apple-Authorized Service Provider or Apple Retail Store for evaluation, and replacement if necessary. You may be eligible for a replacement adapter free of charge provided there are no signs of abuse. Make a reservation at the Genius Bar before visiting your local Apple Retail Store. To make a reservation, go to http://www.apple.com/retail/geniusbar/.

Apple also notes that users should also contact the service provider by phone, before ringing the bell on the Genius Bar desk, "to confirm that the service [they] need is offered at the location." Seemingly, not all Apple Stores will be able to replace broken MagSafe adapters.

Also note that it's probably useless to try and dupe Apple Store employees, telling them your MagSafe spontaneously ignited, or shed its isolation, if you know that's not "entirely" true. They'll probably have one good look at it and know immediately it was the dog who chewed on it. The Genius at your local Apple Store has "extensive knowledge of Apple products and can answer all your technical questions." But whether it's troubleshooting problems that you are having, or need actual repairs to your hardware, you must always remember to make an appointment. Your "journey" towards having your MagSafe replaced begins here.