Ubuntu – Brand new battery is 100% charged but at 0% capacity

I just bought a brand new battery to replace one that had seemingly died. I followed the instructions that came with the battery and charged it overnight, then ran my computer on battery until it was in a critical state and charged it overnight again. Now the battery appears to be 100% charged and at the same time at 0% capacity—when it's plugged in it doesn't charge and when it's unplugged the computer immediately goes into low power mode.

I've taken the battery out and checked the contacts. I've rebooted the system with and without the battery. Nothing seems to help.

I'm not sure if I should blame the battery, the laptop, the charger, or the power management software for this state of affairs. The obvious answer would be the battery but I bought this battery because another year-old battery exhibited similar behavior—I assumed it was dead and a Dell tech came to the same conclusion. I'm loathe to order another new battery without first exploring other explanations.

What can I do to diagnose this problem?

Here's the /proc info:

$ cat /proc/acpi/battery/BAT0/state
present:                 yes
capacity state:          critical
charging state:          charged
present rate:            1 mA
remaining capacity:      0 mAh
present voltage:         9754 mV

$ cat /proc/acpi/battery/BAT0/info
present:                 yes
design capacity:         6600 mAh
last full capacity:      6789 mAh
battery technology:      rechargeable
design voltage:          11100 mV
design capacity warning: 660 mAh
design capacity low:     200 mAh
cycle count:          0
capacity granularity 1:  66 mAh
capacity granularity 2:  66 mAh
model number:            DELL WK
serial number:           4043
battery type:            LION
OEM info:                SMP

[UPDATE] It turned out to be the AC adapter. I find it very strange that a faulty adapter can cause the charging state to be "charged" on an empty battery.

I'll still up-vote answers that give good debugging strategies for this situation—swapping out the battery and adapter, in turn, is obvious but one doesn't always have a spare battery or adapter around to play with. As noted below, Dell was actually prepared to send a technician to my house to replace the motherboard, which would have been a phenomenal waste of time and money.