MacBook Pro battery will not charge even though cycle count is below max count
It appears that your battery is dead or there may be an issue with a component in the charging system. Only a trip to Apple will resolve this.

I had a similar issue with my similar MacBook Pro. The battery showed bad, and it would only run when plugged in. It turned out to be a problem with the charge indicator unit (the thing that shows 5 tiny lights). It had a bad chip, and it was not allowing the battery to charge. A quick swap of the part and my Macbook Pro was back to normal, showing a full battery.

Send to Apple or go to the Apple store, as these kinds of things often show in very strange ways.