I own a 1.25G eMac serial number G8417479QQH
Purchased 4/24/04.

Same problem.

another.

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I was able to get Apple to fix this for me. I contacted the Department of Consumer Affairs, and they sent my complaint to Apple. Within 2 weeks of filing my grievance with the DCA, I had a call from Apple's Office of Executive Relations, and they authorized the free replacement of my logic board.

I live in New York.

I've had the same problem and am embarrassed to say it has taken me a LONG time to look into fixing the eMac, since I've got a couple other computers I can use. But I didn't want to use it as a "doorstop," as Apple Support suggested, when I called to see if they would repair it! I thought about replacing the capacitors, but will try contacting DCA, first. Thanks for the idea. I'll let you know what happens.
#G84296R6QJ8

Good luck!