Chosen Solution
Just replaced the battery (easy - great kit! from IFixIt). However, after powering my computer back up I noticed the magsafe light going from orange to green, then to orange… repeats endlessly. System Information says the battery is not charging. I tried resetting PRAM and SMC, but I keep getting the same problem. I opened the computer again, disconnected the battery, connected it again, same problem… tried resetting PRAM and SMC again… the problem wouldn’t go away. Installed my old battery and the thing runs Ok (only runs out of battery in about 2 hours). I’m quite disappointed. It seems I just threw my money away.
No need to despair for thrown away money, just a faulty replacement battery obviously, contact the customer service describing the issue as you did here, they’ll send a replacement for sure.