Promoting what What Mayer Said; That sounds like a Power Supply to me.
A few things to try:
- Hook an external screen to your iMac! When your iMac goes black, if they both turn off, it's graphics card or power.
- Listen for the fans, feel for air flow out of the top; if you can confirm that it's Geniunely OFF, then you're lookin' at power supply.
PS...let us know if you do replace the battery! One more step for science!