To make sure that poor contact is not causing the issue, since you are positive that the battery starts out good. Remove it from the camera and clean the metal battery contacts inside of the battery compartment and on the battery with a q-tip and alcohol. The connection is very critical and a very small amount of residue or dust can inhibit the power transfer. Also, check how well the battery door shuts and if necessary adjust the metal contacts by gently prying them with a small screw driver to increase the tension.
Also, if you have not already done so, try to power the camera on without the memory card inserted. If the camera comes on without the card, please let me know.
Please keep in mind that my diagnosis & solutions provided are directly dependent on the accuracy of your description of the problem. As with any "do it yourself" fixes, success is a "team effort", since I can't see or touch the camera, and relies on the customer's manual dexterity and ability to follow the instructions well.
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