The battery is replaceable. Hopefully some aftermarket chargers will come out soon so the battery can be charged out of the camera. That way you can use the camera while the other battery is charging. The battery is a Sony NP-BX1.
I highly recommend getting the Wasabi battery kit for this and any other camera you have. I discovered the Wasabi kits, for sale on Amazon, when I bought a Panny GF-3 as a gift for a friend. They are available with one or two batteries (I choose two), lower price than Sony brand, come with charger, car adapter, and round plug adapter (great for travel). Finally, the Wasabi replacement batteries have a higher capacity than the original Sony or Panasonic that I've used (including the RX100). Extra battery, and you can charge without tying up the camera in the mean time.
When I got my camera in Aug 2012, I was not willing to put up with having to charge the removable battery in the camera. So I bought a kit that had a charger and two batteries. It wasn't the Wasabi one mentioned below. My batteries are Power 2000 3.6V, 1600mAH batteries. At one point someone warned me that using off-brand batteries could damage the camera, but I've had no problems at all and get good camera life from them. I found some pill boxes at Container Store that are a good size for these batteries. I've put a couple of small strips of paper criss-crossed in each box wrapping around the battery. I wrote "charged" and "dead" on the ends of one strip where they overlap. I always have an extra battery with me with the "charged" note showing. When I replace a battery, I put the used battery back into the case with the "dead" notation on top. With this system, I won't accidentally put an exhausted battery in my bag; and with three batteries, I can always have two live ones with me while the third is charging.