MacBook’s battery “Not Charging”

As suggested, SMC reset should be your first port of call.

If that doesn't change anything, try removing the battery and see if the system recognises the change. (Status should change to "no battery"/battery icon with an x). If it changes, that pretty much means that the battery is toast.

I'm guessing that your battery has failed as this is the most likely cause of these symptoms. Cycle counts aren't a sure fire way of determining battery health, they can just die.

If your battery is less than 12 months old, you are entitled to a replacement from Apple.

Yes - the SMC reset should resolve any temporary issues with the hardware not sensing the charge adapter. It's even called out in this KB article by Apple as something SMC reset can potentially address:

http://support.apple.com/kb/ht3964

Once you ruled out SMC needing to be reset, you should get the Mac in to a repair shop or do what they would do to isolate the hardware failure:

• test your adapter with another computer
• test your computer with another adapter
• test your computer with another OS (last resort - SMC and the charging behavior you report can be tested with the OS powered off and only the firmware / SMC regulating the LED color and charge state based on the data it can collect which is faulty in your case)

Once they have isolated the failure (likely to your computer), they would open it and inspect for any corrosion or damage on the logic board and charging circuitry and the either swap and test the magsafe board or the battery or the logic board itself based on experience and the hardware service manual for your model.