What is the problem with my NOKIA 1600 mobile?

It is around one and a half year since I purchased it. Now, the battery is causing some trouble. I think it's capacitor is not working properly as it gets discharged very quickly, sometimes even 5 min after charging to full charge. Again when I recharge it, after around 2-3 min it will show "Battery Full" and this goes on. Sometimes, when I dial a number I noticed enough charge was there but while I am talking it suddenly beeps, displays "Battery Low" and gets switched off. This is really irritating. Can you identify the cause of the problem? Should I replace the battery? What do you suggest?

For the specifications see this link: http://www.fonearena.com/nokia-1600_254.html