Why does my computer shut off after 1 second?

It's definitely premature to assume the logic board is bad and spend hundreds of dollars without further testing.

I would pull the battery and only do testing on AC power. Also, try re-seating the RAM, although that is not likely the issue.

I would reset the PMU/SMC and PRAM. If that does not help, try holding down the option key immediately when you power on, and see if it takes you to the option menu without shutting down. If it does, that's a good sign. You can try booting into safe mode by holding down the shift key while powering up. You can try booting from the original OS media, and see if it will let you repair/reinstall your OS. You can connect your machine to another Mac via Firewire cable, boot your Mac in target mode, and see if your hard drive shows up on the other Mac so that you can back up your data. You can boot the other mac in target mode, boot yours in option mode, and then select the other computer's hard drive and boot up as that machine. If that works, this is a strong indication that your hard drive has a problem or needs to be wiped.

Also, I may be getting ahead of myself -- by "as soon as the computer boots", do you mean as soon as the OS loads, or do you mean as soon as the screen comes on? If you can please tell me that, it will let me know how to proceed. The large paragraph above was written assuming the computer dies immediately after the screen lights up.