If it immediately turn off when you disconnect the charger and go on the battery, it means that the battery IS dead.
Get a new battery.
Make sure that it's realy discharged before recharging: The Li-ion battery have a limited number of charging cycles. If you use it 1 minute on the battery and recharge, it count as a full cycle.
NEVER recharge the battery if it's charge is over 25%.
When the battery is completely discharged, recharge it IMMETIATELY! Leaving your battery totaly discharged for more than a few hours WILL KILL IT. It's beter to recharge it when it's down to less than 5%.
Electro
October 2012