When did the stone break? Did you start with the stone on a cold grill so they warm up together or did you put the stone on a hot grill? Did the break happen before or after adding the pizza?
The sparse instructions that came with Weber Crafted Glazed Pizza Stone say to place the stone in the grill before starting the grill. That stone goes up to 750 degrees, and you can put a frozen pizza on it. It is thicker than the thin pizza stones that we have. We used our new Weber stone once so far, to bake a frozen pizza. It did not crack. The pizza came out okay.
I once tried a thin baking stone on our older gas grill and it broke apart the same as yours. I heated the grill and stone up to the full 750 or so. That worked fine, until I added the fresh pizza to it. As soon as the pizza hit the stone, the stone cracked apart.
If I were buying a replacement stone for your grill, I would make sure it is made of thick cordierite. A thick stone means more time to warm up, but it should do better with thermal shock. Cordierite is the most common baking stone material. Companies choose cordierite because it has the property of not expanding or contracting much in different termperatures, so it is less likely to crack in use.