Dell boots and then shuts down

In my office we have two Dell computers. The one that my bookkeeper uses began having a problem this morning. When the computer is turned-on, it goes through the normal boot process until the desk top is shown, and then simple shuts its self off. It is not rebooting, it is completely turned off.

I have disconected the DSL line and it does it. It booted in SAFE mode and it does it. The fan is running and blowing air out of the back. The computer starts very normally, until it shuts off. I has run long enough for me to shut down and restart, but once it has restarted it shuts itself off.

Any suggestions on how to fix this?