I had the same problem and found this thread on Google.
Mine was stuck at exactly 43 % as well. Guessed that is to close to be a coincidence and opened the battery. Since mine is about 3 years old, guarantee on the battery is long gone and I had nothing to loose. But I still like my VAIO and the batteries had about 10% wear only.
The solution was really simple: One cell lead has broken from the central PCB. Soldering it back together fixed the problem instantly. It looks like the cells were once glued to the case, but they were no longer fixed in there and might have moved a bit during transport which broke one of the leads... they don't look very sturdy...
There are 4 cells in there, two larger and two smaller ones. The cells with the same size are linked in series and then linked in parallel for larger capacity. In my case one of the bigger cells was the issue, therefore the 43% remaining capacity of the smaller cells connected only. Guess something around 57 % would be an indication for the smaller circuit broken...
To open the external battery, there are some screws, two hidden below the caps close to the fixed metal clip that holds the battery to the laptop at the back. But even with them removed it needs some force and scissors to cut the really sticky glue to separate the three parts of the case. Make sure to apply as little force as possible on the cells to not damage them. I don't know why they used so much glue, the case snaps nicely together and is stable with all screws back in place.