It would be a more serious issue, if the computer is plugged in, it does not matter if the battery even completely removed, it should not power itself off.

The Left I/o board connects all the power components together. Could be a problem with that.

Did you try resetting the SMC? The SMC could affect the following:

Fans

The computer's fans speed up quickly and remain at high speed.

The computer's fans run at high speed although the computer is not experiencing heavy usage and is properly ventilated.

Lights

The keyboard backlight appears to behave incorrectly (on Mac computers that have this feature).

The Status Indicator Light (SIL) appears to behave incorrectly (on Mac computers that have an SIL).

Battery indicator lights, if present, appear to behave incorrectly (on portables that use non-removable batteries).

The display backlight doesn't respond correctly to ambient light changes on Mac computers that have this feature.

Power

The computer doesn't respond to the power button when pressed.

A portable Mac doesn't appear to respond properly when you close or open the lid.

The computer sleeps or shuts down unexpectedly.

The battery does not appear to be charging properly.

The MagSafe power adaptor LED doesn't appear to indicate the correct activity.

System Performance

The computer is running unusually slowly although it is not experiencing abnormally high CPU utilization.

Video

A computer that supports target display mode does not switch into or out of target display mode as expected.

A computer that supports target display mode switches into or out of target display mode at unexpected times.

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http://support.apple.com/kb/HT3964?viewl...]

if this is happening while you are plugged into AC, then I'd say it's unlikely to be the battery alone.

about the battery itself - select About This Mac in the Apple menu, then click on More Info, this will run System Profiler. On the left side, under the Hardware section, select "Power" and you can check info about your battery's condition on the right side of the window. for example see what it says beside Condition and # of cycles. more than about 300 cycles is considered the normal lifespan of the battery, according to Apple. a dying battery may report its condition as "Check battery". with my MBP, over 3 yrs I've had two batteries die prematurely (<300 cycles) - in one case, when running on battery power, the machine would suddenly shut down without warning at 40% remaining charge. AppleCare replaced both my batteries.

but I've not had a shutdown while plugged in like you report. I'd guess you may have some other issue going on, even if your battery is dead/dying. perhaps a flakey AC adapter?? do you have a friend with a MBP AC adapter that you can borrow to try?

Maybe the heatsinks are dirty from age

Try cleaning the fans and heatsinks

My Battery completed 931 cycles, my MBP 13 unibody, shut down when the cord is plugged in or out. I'm looking for a new battery for replace it and I hope the problem is gone...

If you know about what percent of battery life the computer shuts down, you should check out low battery saver. It's an app that makes sure your Mac sleeps before the battery completely runs out and lets you set a warning to alert you that the computer is about to sleep. (http://itunes.apple.com/us/app/low-batte...) It won't fix your battery problems but at least you won't lose your work, have to restart your computer etc.

Update

Excuse me, low battery saver.