Is this just a dead battery...?

Question

…or is it something more serious?
Here's the situation: I keep my 5-year-old 13" white Macbook (macbook2,1) plugged in most of the time, so the battery doesn't get a lot of exercise, and consistently reports fully charged. Lately, when I use it on battery, it starts up OK, or seems to — and reports fully charged — but as soon as I do anything interesting (e.g., open Safari) it just shuts off. Cold. And won't restart (it'll chime, but that's all), until I plug it in; the battery still reports full charged, however. If I leave it plugged in awhile, it'll start on battery, but as soon as I load a web page...
I've tried taking out the battery, starting up plugged in and then replacing it, hoping that might reset something. No luck.
Is this simply a dead battery? It acts like it's "not holding a charge", but the system, suspiciously enough, still reports fully charged, or nearly so. Before I bounce for a buck and a quarter for a replacement battery only to find I still have a problem, I thought I'd put the question up before the real geniuses...
Is this simply a matter of a failed battery, or is there more to this than meets the ocular organ?
TIA, points will be awarded...
Jeff

Answer 1

OK, the above was the report after the MacBook had been plugged in for a few days. I decided to see what came up when I started on the battery, but the first time I tried it, as soon as Safari launched (it's a login item) it went down. So I let the battery get back up to 100%, and restarted, bypassing login items. System Info reported this:
Battery Information:
  Model Information:
  Manufacturer:          DP
  Device Name:          ASMB013
  Pack Lot Code:          1
  PCB Lot Code:          0
  Firmware Version:          102
  Hardware Revision:          300
  Cell Revision:          200
  Charge Information:
  Charge Remaining (mAh):          4120
  Fully Charged:          Yes
  Charging:          No
  Full Charge Capacity (mAh):          4273
  Health Information:
  Cycle Count:          37
  Condition:          Normal
  Battery Installed:          Yes
  Amperage (mA):          -2054
  Voltage (mV):          10705
I note that those last two items are quite different, and indeed, the voltage had been reduced to < 11V. Dunno if that's significant.
I then tried launching Safari and it promptly went down. It occurred to me that the issue might involve Safari specifically in some way, so I let the battery charge back up again, de-logined Safari, and restarted. I was able to launch a few smaller applications; then I launched Backup, so far so good, but when I tried actually starting a backup, poof.
It's too bad I can't either find a MacBook to borrow to try with my battery or borrow a battery; no one I know has an appropriate model MacBook.