Macbook won't run off charged battery

Question

My battery indicators on screen and underneath the Macbook show maximum charge but when I remove the power adaptor, it dies.
This started to happen a few days ago after a couple of years with zero problems. I tried re-setting the SMC but it makes no difference. As the battery is charged, it seems unlikely just changing it will help.
Any ideas please?
The spec is.
MacBook 3,1
Intel Core 2 Duo
2 Ghz
Serial Number W8****2
<Edited by Host>

Answer 1

Hi - here is the full information from System Profiler.
Battery Information:
  Model Information:
  Serial Number:    DP-AS***9ec
  Manufacturer:    DP
  Device name:    ASMB016
  Pack Lot Code:    0002
  PCB Lot Code:    0000
  Firmware Version:    0110
  Hardware Revision:    0500
  Cell Revision:    0102
  Charge Information:
  Charge remaining (mAh):    4574
  Fully charged:    Yes
  Charging:    No
  Full charge capacity (mAh):    4619
  Health Information:
  Cycle count:    350
  Condition:    Good
  Battery Installed:    Yes
  Amperage (mA):    0
  Voltage (mV):    12497
System Power Settings:
  AC Power:
  System Sleep Timer (Minutes):    10
  Disk Sleep Timer (Minutes):    10
  Display Sleep Timer (Minutes):    10
  Automatic Restart On Power Loss:    No
  Wake On AC Change:    No
  Wake On Clamshell Open:    Yes
  Wake On LAN:    Yes
  Display Sleep Uses Dim:    Yes
  Battery Power:
  System Sleep Timer (Minutes):    60
  Disk Sleep Timer (Minutes):    10
  Display Sleep Timer (Minutes):    60
  Wake On AC Change:    No
  Wake On Clamshell Open:    Yes
  Display Sleep Uses Dim:    Yes
  Reduce Brightness:    Yes
Hardware Configuration:
  UPS Installed:    No
AC Charger Information:
  Connected:    Yes
  Charging:    No
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